sum of reciprocals of powers of 2. 2. Theorem A is an imm
sum of reciprocals of powers of 2 For more information about Eneström numbers, or to read E 41 itself in either its original Latin or in translation, visit The Euler Archive. Power (Psychology) Formulas for sums of powers of integers and their reciprocals Authors: Levent Kargin Ayhan Dil Akdeniz University Mümün Can Abstract This paper gives new explicit formulas. Trick to solve sum of the reciprocals of two numbers#shortvideo #shortfeed #maths #tricks There are only four integers equal to the sum of the factorials of their digits. Such numbers are called factorions . times the reciprocal of the denominator. Likewise, write down the formula for series resistors and remember Power is I 2 R. , 74, 2019] considered truncates of the power series of , i. Separate into separate infinite series's so we can calculate each and find the original sum: The first infinite sequence shall be all the reciprocals of the powers of , the second shall be reciprocals of the powers of , and the third will consist of reciprocals of the powers of . So, the reciprocal of -14 is -1/14 Let A(−4,0) and B(4,0). So our equation is going to be minus X squared plus three X plus one equals zero and just identifying the summon product formulas, the some formula is taking the ratio coefficients as minus B over a and the product ratio of the product born you, er is staying … 1 INTRODUCTION. (or) Use the formula, x = 1/x, Here, x = 2/3 Thus, x = 1 / x = 1 / (2/3) = 3/2 Therefore, the reciprocal of a fraction 2/3 is 3/2. 00 Divisors of N are {1, 2, 3, 6} Sum of inverse of divisors is equal to (1/1 + 1/2 + 1/3 + 1/6) = 2. Step 3: Click on " Reset " to clear the field and enter a new fraction. · a. reciprocal() is a mathematical function that is used to calculate reciprocal of all the elements in the input array. 644934. Figure 1 Marby’s visual proof that the sum of reciprocals of even … The sum of a number and twice its reciprocal is 27/5. This article, or a section of it, needs explaining. Write a Python program to sum recursion lists. b. x2 - 2x + 1 = 0. Do mathematic tasks. Obligation a. Pure Appl. Sum of Reciprocals of Even Powers of Odd Integers - ProofWiki Sum of Reciprocals of Even Powers of Odd Integers Contents 1 Theorem 1. 1 INTRODUCTION. Now well sum the so called harmonic series of all reciprocals of the positive from PROB 703 at Sherando High 1 Answer The smallest sum of a number n and its reciprocal 1n is 2 which occurs when n=1 . Let A(−4,0) and B(4,0). So our equation is going to be minus X squared plus three X plus one equals zero and just identifying the summon product formulas, the some formula is taking the ratio coefficients as minus B over a and the product ratio of the product born you, er is staying … Let A(−4,0) and B(4,0). So, 1/x + 1/y = (x+y)/xy, which means denominator must be multiple of two prime numbers. The control of microgrids is operating in different levels of a hierarchical control approach … 1 INTRODUCTION. Reciprocal Reciprocal The reciprocal is simply: 1/number To get the reciprocal of a number, we divide 1 by the number. How to Use the Reciprocal Calculator? Step 1: Enter the numerator and denominator in the respective input boxes. 10/12 = 5/6 = 5/3*2. One number is 2 times another. The control of microgrids is operating in different levels of a hierarchical control approach … This product is no coincidence. n. Consider the sum of reciprocals of powers of 3: N 1 1 1 1 = + + 3i 1 3 3² 3N i=0 Write a recursive function that takes as input N>=0, and then computes and returns the sum above. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the … The Sum of the Reciprocals of Primes: 1/2+1/3+1/5+1/7+1/11+. " Equivalently, We may rewrite any fraction as the numerator. You’ll want to isolate the x ⁵. – Frank Yellin. There is one really fast option. 0:12 Formulas for sine Get the Best Homework solution If you want to get the best homework answers, you need to … Sum of inverse powers of 2. Much of this topic was developed during the seventeenth century. finite sum of distinct terms taken from Hn if and only if for all ε > 0, there is a finite sum s of distinct terms taken from Hn such that 0 ^ s — pjq < ε. Proof 4 ∞ ∑ n = 1 1 (2n − 1)2 = ∫1 0∫1 0 1 1 − x2y2dxdy Sum of Reciprocals of Squares of Odd Integers as Double Integral The sum of a number and its reciprocal is `2(1)/(30 The sum of the number and its reciprocal is 2. In 1735 Euler proved that the sum of the reciprocal. Let n be any integer and let H n be the sum of the first n reciprocals. Oct 24, 2020 at 6:02. CDS Maths Previous Paper 11 (Held On: 7 Feb 2021) Attempt Online. The task is to find the sum of the inverse of the divisors of N. The reciprocal of 10/13 is 13/10. powers of positive . ] The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"): The reciprocals of powers of any n>1 produce a convergent series: Alternating the signs of reciprocals of powers of 2 also produces a convergent series: Alternating the signs of reciprocals of powers of any n>1 produces a convergent series: (Davenport 1980, p. i code the sum of reciprocal of n natural numbers not the squares – shemayon soloman. Corollary Let n ∈ Z > 0 be a (strictly) positive integer . Three-fourths and four-thirds are reciprocals. k =1. 1 Corollary 2 Proof 3 Sources Theorem Let n ∈ Z > 0 be a (strictly) positive integer . In the following example, you'll see how to add and subtract dates by entering positive or negative numbers. 1,158 Solution 1. 34. Or does it diverge and doesn't have a single sum? Edit: I found a calculator to do it. The first few values of such that is prime are 1, 2, 4, 6, 12, 14, 60, 64, 96, 100, . Euler found the exact sum to be / and announced this discovery in 1735. 2. Euler found this … This JAVA program is to find power of a number using recursion. So our equation is going to be minus X squared plus three X plus one equals zero and just identifying the summon product formulas, the some formula is taking the ratio coefficients as minus B over a and the product ratio of the product born you, er is staying … 1. ( 2) ( 1) + = + + + − + = = … 1 INTRODUCTION. Theorem A is an immediate consequence of a result of the author [2, Theorem 4] together with the fact that every sufficiently large integer is the sum of distinct nth. " Quart. The only advantage is that the sum is expressible as Proof 2 Let n be a positive integer . Here we extend Gessel's observations to arbitrary ``thinned exponential series''. \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }\) \(=\) \(\ds \dfrac 1 {1^{2 n} } + \dfrac 1 {3^{2 n} } + \dfrac 1 {5^{2 n} } + \dfrac 1 {7^{2 n . Modified yesterday. Euler’s astonishingly clever method “ has fascinated mathematicians ever since . that is, that the sum of reciprocals of even powers of 2 is \frac {1} {3}, as shown in Figure 1. We have: We also have: So we can deduce: ∞ ∑ n = 0 1 (2n + 1)2 = π2 8 Proof 3 This "proof" leads to a dead end. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, . Besides, the sum of reciprocals of po we rs of the first n natural numbers. 6)0 18,26)0 18-22) Answers: 1. Hoey listed sums of distinct factorials which give square numbers, and J. So formula (6) implies the formula for the sum of reciprocal squares . Proof 1; Proof 2; Proof 1 Write the numbers in base 2: The powers of 2 starting from 1 = 2 0 will be in binary, 1 + 10 + 100 + 1000 will always be a number that will be a n with all binary digits 1. Example: Find the reciprocal of 2 / 3. 0654<R<2. Example. If R{\displaystyle R}denotes the maximum sum of reciprocals of a sum-free sequence, then through subsequent research it is known that 2. For instance, the sum of the reciprocals of the powers of two (a geometric series) is two. Example: the reciprocal of 2 is ½ (a half) More Examples: Like Turning the Number Upside Down For Fractions, Flip the Whole Fraction Over Example: the reciprocal of 3/4 is 4/3 Read more at Reciprocal of a Fraction. The camera scanning is awesome and really helpful if u r in a hurry to finish a sum ! I really recommend this app to all school and college students. More ways to get app Get calculation help online What is the sum of the reciprocals of the infinite series 2 to the power of squares? If I'm asking that right. Ask Question Asked yesterday. In this paper, two formulas are given for the sum of powers of positive. It goes as follows. 26 Mar. Proof 1; Proof 2; Proof 1 To find the sum of a finite geometric series, use the formula, Sn=a1(1rn)1r,r1 , where n is the number of terms, a1 is the first term and r is the common ratio . $$ Solution 2 "Find" in the sense of "having a closed form expression"? Likely, no, under most reasonable interpretations … To sum columns or rows at the same time, use a formula of the form: =sum(A:B) or =sum(1:2). Proof Sources What is the sum of the reciprocals of the infinite series 2 to the power of squares? If I'm asking that right. Definition: Two numbers whose product is 1 are called reciprocals of each other. That rule in the box is called The Definition of Division. 1. Math. Solution: The sum of the series is approximately equal to 1. Solved Example 2: Find the reciprocal of the integer -14. alternative obligation from reciprocal obligation. It has been possible to obtain the general result for the sums of an arbitrary power of reciprocals of square numbers. Find the reciprocal of 1. We reviewed their content and use your feedback to keep the . Enter the number of days to add or subtract in column B. In the given options we can see that only option B fulfills this. Its value lies in showing that this approach does not work. To learn how to find the reciprocal of different numbers, continue reading. Expressly stipulated to that effect general, creditor can refuse valid tender of payment. The first terms of the series sum to approximately +, where is the natural logarithm and is the Euler–Mascheroni constant. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the … z2 π2n2!. Now well sum the so called harmonic series of all reciprocals of the positive from PROB 703 at Sherando High In other words,we are adding up the reciprocals of the first 100 natural numbers. The multiplication of these two … Sum of inverse powers of 2. Doing homework can help improve grades. While no factorial greater than 1! is a square number, D. Using sigma notation, we can rewrite the summation in the following way: The above summation is read as "the sum of k=1 to 100 of 1 over k squared. 56446841360593857933. J Pi Maths 1 Author by Community Updated on August 12, 2022 Eric Naslund about 11 years There is no known closed form expression. Trick to solve sum of the reciprocals of two numbers#shortvideo #shortfeed #maths #tricks J. Therefore, H n = (even number + odd number)/2 N *odd number . The sum of the reciprocals of two alternate natural numbers is (7/24). Go to the editor Click me to see the sample solution 3. Step 2: Click the button “ Solve ” to find the reciprocal value of the fraction entered. Consider a mixed fraction, 4 (1/2). For example, the sum of the reciprocals of the triangular numbers is easily determined . So, the reciprocal of -14 is -1/14 Equation 1: Sum of the reciprocals of even powers of integer numbers. and provide a general technique for estimating such sums. Hint: Allow the user to repetitively input . What is the sum of the numbers? This question was previously asked in. Therefore, 13/10 is the reciprocal of 10/13. Who are the experts? Experts are tested by Chegg as specialists in their subject area. Any other value of n will produce a larger sum. So our equation is going to be minus X squared plus three X plus one equals zero and just identifying the summon product formulas, the some formula is taking the ratio coefficients as minus B over a and the product ratio of the product born you, er is staying … 2 3 (3) (2) 1 0. Now well sum the so called harmonic series of all reciprocals of the positive from PROB 703 at Sherando High J. You can enter a negative number to subtract days from your start date, and a positive number to add to your date. nyu langone urgent care long island. = 1. Division is defined in terms of multiplication, just as subtraction can be defined in terms of addition: a − b = a . Such series appear in many areas of modern mathematics. 25, 347-362 . This is the largest number having that many digits. The sum of a number and its reciprocal is `2(1)/(30 The sum of the number and its reciprocal is 2. [1] Density[edit] One way to interpret this fact is that there must be a “lot” of primes—well, of course there are an infinite number of them, but not every infinite set of natural numbers has a reciprocal … Question: One number is 2 times another. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it … The first terms of the series sum to approximately +, where is the natural logarithm and is the Euler–Mascheroni constant. Trick to solve sum of the reciprocals of two numbers#shortvideo #shortfeed #maths #tricks Now well sum the so called harmonic series of all reciprocals of the positive from PROB 703 at Sherando High 2 3 (3) (2) 1 0. Let be the number of ways an integer can be written as a sum of two or more consecutive primes. Select one: No liability for fortuitous events intervene. Sum of fourth powers of reciprocal natural numbers1;2 c Mark Loewe, Austin, 2 May 2020 Euler1 divides the in nite power series sin = 1! 3 3! + 5 5! 7 7! + (1) by y= sin to obtain 0 = 1 1!y + 3 . Now 1/3+1/2 = 5/6. Write a Python program to calculate the sum of a list of numbers. We can easily calculate these to be respectively. Step 4: Solve. Sums of Reciprocal Powers You have seen that the harmonic series diverges. Find the numbers. Solving systems of linear equations: tables represent two linear functions in a systemöwhat is the solution to this system? 0 (10)0 (1. onenote conflicting changes are highlighted in red. 577215. If the number of points on the circle x2+y2 =16 such that the area of the triangle whose vertices are A,B and C is a positive integer, is N then the value of [N /7] is, where N represents greatest integer function. Remember that you can also use the keyboard shortcuts CTRL + Figure out mathematic questions; Do mathematic equation; Get help from expert tutors; Solve Now! Use AutoSum to sum numbers. " In otherwords, we are adding up the squares of the reciprocals of the first 100 natural numbers. SO it is of the form 2 n + 1 − 1 – P … Factorial Sums Download Wolfram Notebook The sum-of-factorial powers function is defined by (1) For , (2) (3) (4) where is the exponential integral, (OEIS A091725 ), is the E n -function , is the real part of , and i is the imaginary number. In particular: I believe only "if" is true as shown in Product Form of Sum on Completely Multiplicative Function . [E 41 is the Eneström number of Euler's De Summis Serierum Reciprocarum. The sum of fourth powers of reciprocal natural numbers is 16=15 times Eq (13),2 X1 n=1;2;3; 1 n4 = 1 14 + 1 24 + 1 34 + 1 44 + = 16 15 15 16 1 14 + 1 . Expert Answer. There is the famous sophmore's dream identity: $$\sum_{k=1}^{\infty} \frac{1}{k^k} = \int_0^1 \frac{dx}{x^x}. k p = 1 + 1. Partial fulfillment Business Accounting Q&A Library In a reciprocal obligation, the remedy of the injured party is: Alternative Cumulative None of the above. Today we are going to be finding the sum and product of the roots without solving the equation. The easiest thing is enumerate all the possibilities (few seconds of CPU). Viewed 30 times 0 I have one table with a value column, and supply start and supply end columns, and a date table I wish to use to filter it, specifically using a start of month . We can find the reciprocal of a nonzero number in fractional form by inverting it (that is, by switching positions of the numerator and denominator). The Total Power will be V 2 /R p so multiply both sides of the parallel resistor equation by V 2 and that produces what you want. For example if base is 2 and exponent is 3 then the power of a number is 23 = 8. . (1) 0. Go to the editor Click me to see the sample solution 2. Get the Best Homework solution Sum of reciprocals of powers; Sum of reciprocals of powers. We consider . Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it … It goes as follows. Examples : Input : n = 30 Output : 72 Dividers sum 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72 Input : n . ( ) 1 2 3 . … J. The only advantage is that the sum is expressible as The reciprocal of a fraction can be found by interchanging the numerator and the denominator values. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. The numpy. 2 days ago · Power BI: SUM values if dates between two columns lie within month selected by date slicer, with prorated value. Step 3: Convert the decimal to fraction. (6) Now Euler proposes to actually perform the multiplication on the right hand side and compare with the power series (2). The dimensions of a metallic cuboid are 100 cm×80 cm×64 cm. Solution: Step 1: Make it a numerator by placing one in the fraction. Answers: 2 Show answers . Write a Python program to convert an integer to a string in any base. Despite the divergence of the sum of reciprocal primes, the alternating series (7) (OEIS A078437) converges . Pure Obligation- one whose effectivity or extinguishment . Reciprocals come in pairs. To verify the answer, we will multiply 10/13 with its reciprocal and check if the product is 1 (10/13) × (13/10) = 1. 8570}. The control of microgrids is operating in different levels of a hierarchical control approach … An infinite sum of the form (1) is known as an infinite series. The control of microgrids is operating in different levels of a hierarchical control approach … SUM OF THE RECIPROCAL INTEGERS It is a simple matter to sum up the sum of the first N integers. The term with z is certainly 1. 1/2 + 1/4 + 1/8 + 1/16 + . McCranie gave the one additional sum less than : (29) The reciprocal of x = 1/x Alternatively, you can say that if you multiply a number by its reciprocal, you'll get 1: For example, if our chosen number is 5, its reciprocal is 1/5. (x - 1)2 = 0 Find an Online Tutor Now. You can solve it more to find the … Given a number n, the task is to find the sum of all the factors. 8570{\displaystyle 2. For . In this case, Article 1174 shall apply. Sum of inverse powers of 2. Also, let 2 N be the largest power of 2 that is less than n. Timely Delivery. … Let A(−4,0) and B(4,0). In cell C2, enter =A2+B2, and copy down as needed. The … Trick to solve sum of the reciprocals of two numbers#shortvideo #shortfeed #maths #tricks. So our equation is going to be minus X squared plus three X plus one equals zero and just identifying the summon product formulas, the some formula is taking the ratio coefficients as minus B over a and the product ratio of the product born you, er is staying … In other words,we are adding up the reciprocals of the first 100 natural numbers. Solution 3. Leonhard Euler continued this study and in the process solved many important problems. And again, it's unclear whether you're adding the sum of the reciprocal (your title) or the sum of the square of the reciprocal (your code). Examples: Input: N = 6, Sum = 12 Output: 2. 0 Input: N = 9, Sum = 13 Output: 1. (Davenport 1980, p. 422784. The control of microgrids is operating in different levels of a hierarchical control approach … 1 Answer The smallest sum of a number n and its reciprocal 1n is 2 which occurs when n=1 . Step 2: Swap the numerator and denominator. The control of microgrids is operating in different levels of a hierarchical control approach … Explanation: consider this product, (1 + 1/2 + 1/4 + 1/8 + 1/16 …) (1 + 1/3 + 1/9 + 1/27 + 1/81 …) (1 + 1/5 + 1/25 + 1/125 + 1/625 …), so as you see, this product gives the reciprocal of any number that contains only 2 and/or 3 and/or 5 in their prime factorization. e. find the number. 44 Recommended: Please try your approach on {IDE} first, before moving on to the solution. 34). corresponds to. A microgrid consists of a set of distributed generators (DGs), renewable energy sources (RESs), energy storage sources (ESSs), and interconnected loads, that can be operated in islanded or grid-connected modes [1, 2]. Solution: The number -14 can also be written as -14/1. How Euler found the sum of reciprocal squares A. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it … The reciprocal of 10/13 is 13/10. 2: Sum and Difference Identities How to use the Sum and Difference Formulas to find the exact value of the sine, cosine or tangent of an angle. = − = −= = − = =− =− ψ γ ψ γ ψ γ ψ γ In a way, the above formula for ψ(x) is really no simpler computationally then just summing up the terms 1/n since , in getting ψ(x), one is really carrying out the same type of sum calculations . Reciprocal of decimal numbers: It is easy to find the reciprocal of numbers and integers but when they have a decimal point, it is a little different. Oct 24, 2020 at 6:18. Although there are series related to … Here we have to find the number which is sum of reciprocals of two prime numbers say x ,y. 7. Now well sum the so called harmonic series of all reciprocals of the positive from PROB 703 at Sherando High alternative obligation from reciprocal obligation This is a single blog caption. Thus must be equal to −z3/6 in (2), so X∞ n=1 1 π2n2 = 1 6. Whether you're struggling with a difficult concept or just need someone to bounce ideas off of, expert professors can be a great . Trick to solve sum of the reciprocals of two numbers#shortvideo #shortfeed #maths #tricks Today we are going to be finding the sum and product of the roots without solving the equation. Outline of Euler's calculation of \(\sum_{n=1}^\infty \frac{1}{n^2}\), with text from E 41. You can use the reciprocal fraction calculator or solve it by yourself. Now well sum the so called harmonic series of all reciprocals of the positive from PROB 703 at Sherando High Today we are going to be finding the sum and product of the roots without solving the equation. sequences-and-series. Eremenko November 5, 2013 In the lectures, the formula X∞ n=1 1 n2 = π2 6 (1) was derived using residues. … Today we are going to be finding the sum and product of the roots without solving the equation. . What about the sum of reciprocal squares? In fact, they converge, and to something very … In other words,we are adding up the reciprocals of the first 100 natural numbers. Reciprocal of a Mixed Fraction In order to find the same for a mixed fraction, convert it into improper fractions and perform the operation. Solution: To find the solution … Question: One number is 2 times another. The first table contains the first custom column titled TrackingHistory [_BeginningFootage], and the second table contains a column titled InventoryTransDetail [_TotalGoodFootage]. Example: the reciprocal of 2 is ½ (a half) More … Bach and Shallit (1996) show that. if and only if $\ds \sum_{n \mathop = 1}^\infty a_n n^{-z}$ is absolutely convergent. The term with z3 is −z X∞ n=1 z2 π2n2. Combin. When you do the whole common denominator routine, every reciprocal except for 1/2 N will end up with an even number as numerator. ” Euler had previously proved the Basel … Solution 3 Separate into separate infinite series's so we can calculate each and find the original sum: The first infinite sequence shall be all the reciprocals of the powers of , the second shall be reciprocals of the powers of , and the third will consist of reciprocals of the powers of . You could follow the procedure described here to find the sum of the reciprocal 4th powers. The sum is . Proof 1; Proof 2; Proof 1 The reciprocal of 2/3 is 3/2. I have two columns I have created in a BI report that are in tables that are unrelated. Question: One number is 2 times another. X. J. His arguments were based on manipulations that were not justified at the time, although he was later proven … In other words,we are adding up the reciprocals of the first 100 natural numbers. View all CDS Papers > 12; 13; 14; 16; Answer (Detailed Solution Below) Option 3 : 14. How to Write the Reciprocal of a Number? Reciprocal Reciprocal The reciprocal is simply: 1/number To get the reciprocal of a number, we divide 1 by the number. It is done as follows- 2 ( 1) [(1 ] [2 ( 1)] . 567 Consultants 93% Recurring customers 90510+ Student Reviews Get Homework Help In other words,we are adding up the reciprocals of the first 100 natural numbers. Go to the editor Test Data: [1, 2, [3,4], [5,6]] Question: One number is 2 times another. There are 1431 ways to extract distinct cards such that the sum of their reciprocals is 1. The number series should go 1/1 + 1/2 + 1/16 + 1/512 + 1/65,536. Trick to solve sum of the reciprocals of two numbers#shortvideo #shortfeed #maths #tricks He developed a general solution to all even-powered reciprocals. "a divided by b is equal to a times the reciprocal of b. Infinite sums of reciprocals of certain figurate numbers are often discussed in a first-year calculus course. 567 Consultants 93% Recurring customers 90510+ Student Reviews Get Homework Help 1 INTRODUCTION. polynomials of the form , and established combinatorially that the reciprocal of the truncate has a power series with all coefficients non-negative precisely when is odd. The sum of their reciprocals is 9 . "On the Sums of the Inverse Powers of the Prime Numbers. So our equation is going to be minus X squared plus three X plus one equals zero and just identifying the summon product formulas, the some formula is taking the ratio coefficients as minus B over a and the product ratio of the product born you, er is staying … A carefully drawn graph show show you why $$ \text{that little bit} < \left( 1 - \frac 1 {\sqrt 2} \right) + \left( \frac 1 {\sqrt 2} - \frac 1 {\sqrt 3} \right) + \left( \frac 1 {\sqrt 3} - \frac 1 {\sqrt 4} \right) + \cdots + \left( \frac 1 {\sqrt{50}} - \frac 1 {\sqrt{51}} \right) $$ and all of the terms cancel out except the first and the . Ideally, I would like to be able to subtract the good footage from … A famous theorem by Leonhard Euler, possibly the greatest, most prolific and creative mathematician ever. An outline is given to extend the result to the general case of the sums. Also, write a main function that calls your recursive function at least once then displays the result using printf. I mean 1/ (2^ (n^2)) + 1/ (2^ ( (n+1)^2)) + .