Infinite series Definition and 390 Threads

Dear friends, \sum_{x=1}^{\infty}\frac{1}{x} diverges. But \sum_{x=1}^{\infty}\frac{1}{x^{2}}=\frac{\pi^{2}}{6} How can we prove that \sum_{x=1}^{\infty}\left(\frac{1}{x^{\left(1+epsilon\right)}}\right) converges to a finite value? Thanks in advance. Bincy.

Homework Statement Consider the infinite series s(x)=1-tan^2(x)+tan^4(x)-tan^6(x)+... , where 0<x<pi/4 s'(x)= A.sin2x B.cos2x C.-tan2x D.-sin2x E.-cos2x The Attempt at a Solution Attempting to derive the series, i get 1-2tanxsec^2x+4tan^3(x)sec^2(x)-6tan^5(x)sec^2(x)+... am i...

Homework Statement Consider the series: \sum\frac{1}{n!}, where n begins at one and grows infinitely larger (Sorry, I'm still a bit new to the equation editor on here :) ) 1) Use the ratio test to prove that this series is convergent. 2) Use the comparison test to show that S < 2 3)...

Homework Statement Ʃ 4/(n(n+2)) from n=1 to n=infinity Homework Equations The Attempt at a Solution I tried using partial fractions to get A/n + B/(n+2), and I solved for A and B to get A=2 and B=-2 I tried summing them up, so everything would cancel except the first & last...

I need a quick reminder that this is (hopefully) true: Let \sum a_n be an infinite series of complex terms which converges but not absolutely. Then can we still break it up into its real and imaginary parts? \sum a_n = \sum x_n + i\sum y_n

Show that $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \arctan \left( \frac{1}{2n+1} \right) = \arctan \Bigg( \text{tanh} \Big( \frac{\pi}{4} \Big) \Bigg)$.I'm tempted to give a hint (or two) right off the bat. But I'll wait.

Homework Statement If possible, evaluate the sum : http://www4a.wolframalpha.com/Calculate/MSP/MSP31841a0i89gaa1b8f5c80000373e40eh779f93h7?MSPStoreType=image/gif&s=17&w=109&h=47 Homework Equations The Attempt at a Solution Not really sure what to do. I've tried writing out the...

It looks like you guys love to solve infinite series problems. Here are a few more \( \displaystyle \sum_{n=1}^{\infty}\frac{1}{n^4}\) \( \displaystyle \sum_{n=1}^{\infty}\frac{(n+1) \cdot (n+1)!}{(n+5)!}\)

Is an infinite series of [nonrepeating] random numbers possible? That is, can the term "random" apply to a [nonrepeating] infinite series? It seems to me that Cantor's logic might not allow the operation of [nonrepeating] randomization on a number line approaching infinity.

Homework Statement k is a positive integer. \sum^{\infty}_{n=0} \frac{(n!)^{k+2}*x^{n}}{((k+2)n)!} Homework Equations The Attempt at a Solution I have no idea.. this is too confusing. I tried the ratio test (which is the only way I know how to deal with factorials) but I get...

Homework Statement Determine whether the series converges or diverges 1/2(ln(n+1))^2 from n = 1 to infinity The Attempt at a Solution Cannot find anything to compare this series to that will show it diverges. Ratio and root test both fail. Integral test requires integrating to a...

Homework Statement Investigate the convergence of the following series. (2n)!/(n!n!) The Attempt at a Solution Number one, I don't see how they get that if an = (2n)!, then an+1 = ((2n+2))!, it should be (2n+1)! Number two, I don't see how they go to the second step, why is the second step...

I stumbled upon this infinite series that converges to \pi: 4\sum\frac{\sqrt{n^2-i^2}}{n^2} for i = 1:n as n{\rightarrow∞} I haven't been able to find any similar series online and I'm really curious how to prove this does indeed converge to \pi. Any insight would be greatly appreciated.

Homework Statement Ugh really getting frustrated at not being able to keep up with math.. Find the set of x for which the series converges AND find the sum of the series at these values. s=\sum^{k=infinity}_{k=0} \frac{(x+7)^{k}}{3^{k}} Homework Equations The Attempt at a...

More fun problems! Evaluate the following: 1. \( \displaystyle \sum_{n=0}^{\infty}\frac{(-1)^n \{ (2n+1)\sin^{2n+1}(k)+(-1)^n\cos^{2n+1}(k) \}}{\cos^{2n+1}(k)(2n+1)^2} \) 2. \( \displaystyle \sum_{n=1}^{\infty} \frac{\sin^4(2^n)}{4^n} \) 3. \( \displaystyle \sum_{n=0}^{\infty}\frac{\{...

Question : How do i convert an infinite series into an integral? I searched a few sites and the method given is as follows replace r/n by x repalce1/n by dx replace Ʃ by ∫ which works perfectly fine when i tried a few examples but i don't understand the intuition behind it. Why this...

1 - 1 + 1 -1 + 1 - 1 ... Does that equal 0 or 1? (1-1) + (1-1) + (1-1) + ... = 0 + 0 + 0 = 0 or 1 + (-1 + 1) + (-1+1) + (-1+1) + ... = 1 + 0 + 0 + 0 = 1

Hi, The title of the thread doesn't adequately describe the question I want to ask, so here it is: Suppose we have two infinite series, \sum_{n=1}^{\infty}a_n and \sum_{n=1}^{\infty}b_n, both of which are convergent. Also suppose a_n \leq b_n for all n, and a_n < b_n for at least one n...

infinite series! Homework Statement Hi! I've learned that the definition of a sequence of elements of complex numbers is as follows; a sequence is a function whose domain is a set of all positive integers with values in a set consisting of all complex numbers. (Denote the set of all complex...

I was trying to prove the following but couldn't succeed. Is there a systematic methods to prove that the following infinite sum is positive? (alternating series) sum from n = 0 to ∞ of ((-1)^{n}* x^{n+z}) / (n+z)! conditions x≥0 and z≥1 note: when x≤1, we can directly see that s_{n}-...

I have the infinite series... \sum n^n/n! somehow, I need to use the Ratio Test... and shoe that the resulting limit is equal to e. I can't figure out what I am doing wrong algebraically. Right now I have simplified this limit to lim((n+1)^n/n^n). Please Help... Thanks.

Hi, I'm stuck on a problem because I don't know how to calculate this series: Ʃ(2^n * x^n), n starts from 0 to infinity. How can I calculate this? Do I have to transform it into something else where I know the outcome of the Σ like we do with the geometric functions?

I was doing my math homework when I started thinking about the sums of two infinite series. I determined that the sum of the first series \sum_{n=1}^{\infty} cos(\frac{\pi}{2n}) diverges. I could not figure out whether or not the series \sum_{n=1}^{\infty} sin(\frac{\pi}{2n}) converges or...

Homework Statement Evaluate the sum of the following: \sum\limits^\inf_{n=1} \frac{6}{n(n+1)} Homework Equations The Attempt at a Solution Well... The denominator is going to get infinitely large as n approaches infinity, so would the value of the sum not converge to zero? The...

Sense e^x=Ʃ[k=0,∞] x^k/k! then ln(e^x) = ln(Ʃ[k=0,∞] x^k/k!) x = ln(Ʃ[k=0,∞] x^k/k!) is this true?

Determine how many terms of the convergent series must be summed to be sure that the remainder is less than 10^-4. Ʃ[n=1,∞] cos(k)/k^(3/2) I'm not sure how to solve this problem. I'm only aware of remainder formulas for the integral test and for alternating series. I'm not sure that this...

Determine if the following series converges or diverges: Ʃ[k=1,inf] ( k^2-1 )/( k^3+4 ) I don't see how to solve this problem the divergence test is inconclusive the ratio test is inconclusive the root test is inconclusive the integral test... not sure how to integrate this... the...

Homework Statement Determine if the following series converges or diverges. Ʃ[k=1,inf] tan(k)/(k^2+1) Homework Equations The Attempt at a Solution I have no idea how to solve this problem. Now that I think of it, I have never solved a single question about series were I'm...

Homework Statement Why does this series diverge?Homework Equations \sum_{n}^{\infty }\frac{-1^{(2n+2)}}{n+1}The Attempt at a Solution I must be missing a rule with the -1 sign. My logic is that for all n, the numerator = 1 since anything to the power of 2 is even and adding the 2 doesn't...

This is a similar question that I have to the other one that I posted. Determine if the following series converges or diverges. sum[k=1,inf] 3/(k(k+3)) I applied the limit comparison test with 1/k^2 also k(k+3))=k^2+3k lim k->inf [ 3/(k^2+3k) ]/[ 1/k^2 ] = lim k->inf (3k^2/(k^2+3k)...

Determine weather or not the following series converges or diverges. sum[k=1,inf] 2/(k^2-1) I applied the limit comparison test with 1/k^2 2* lim k->inf [ 1/(k^2-1) ] /(1/k^2) = 2*lim k->inf k^2/(k^2-1) =H 2 then because sum[k=1,inf] 1/k^2 is a convergent p-series then...

State if the following series converges or diverges sum[k=1,inf] cos(k)/(k^2+1) I applied the convergence test lim k->inf cos(k)/(k^2+1) =H lim k->inf -sin(k)/(2k) =H lim k->inf -cos(k)/2 = some undefined value bounded by -1 and 1 =/= 0 so the series diverges by the divergence test. I guess my...

Homework Statement Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10^-5. Although you do not need it, the exact value of the series is given. ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k Homework Equations [b]3...

Homework Statement Find the sequence of partial sums {S_n} and evaluate the limit of {S_n} for the following series .9+.09+.009+... What is .9+.09+.009+... equal to? Homework Equations The Attempt at a Solution For the first part of the question (find the sequence of...

Homework Statement Consider the following convergent series. Then complete parts a throw d below. sum[k=1,inf] 5/k^7 a. Find an upper bound for the remainder in terms of n Homework Equations Estimating Series with Positive Terms Let f be a continuous, positive, decreasing...

Prove that \sum_{i=1}^{\infty }x_{i} = \sum_{i=1}^{\infty }(x_{2i} + x_{2i+1}) if \sum_{i=1}^{\infty }x_{i} converges and if for any \varepsilon > 0 there is some m such that |x_{k}| < \varepsilon for all k\geq m. I'm a little confused by this because for 1/(4^i) for i from 1 to infinity...

Hi, I don't really need help with a problem, just having some troubles understanding something. Find the interval of convergence of the series sigma[n=0,inf] (x-1)^n/n^3 by the root test I got that |x-1|<1 and that 0<x<2 I than have to plug in these values (0 and 2) to see if the...

Homework Statement Prove that sigma[n=0,inf] ((-1)^n n)/(n+1) diverges Homework Equations The Attempt at a Solution I'm unsure how to do this I tried applying the alternating sereis test but when I did so I got (n/(n+1))' = 1/(n+1)^2 so I can't say that the terms are...

Homework Statement Find the Interval of Convergence for the given series. Check the endpoint behavior carefully sigma[n=0,inf] (n (x-2)^n)/( (n+1)4^n ) Homework Equations The Attempt at a Solution I was following along with the answer key and they used the ratio test... The...

Homework Statement Determine whether the following series converges absolutely, converges conditionally or diverges. Show your work in applying any tests used. sigma[k=1,inf] [(-1)^k*k/sqrt(k^4+2)] Homework Equations integral csc(x) dx = -ln|sec(x)+cot(x)| + C csc(x) = 1/sin(x) sec(x) =...

Homework Statement The sereis sigma[k=1,inf] [(-1)^k/k^p] converges conditionally for (a) p<1 (b) 0<p<=1 (c) p>1 (d) p=0 (e) None of the above Homework Equations The Attempt at a Solution The answer key said that (b) was the correct answer and I'm having trouble...

Homework Statement Check out matt grime's post in this thread (it's the last one): https://www.physicsforums.com/showthread.php?p=470773#post470773" How exactly did he know that the sum could be represented as that double integral? Also, is there a method of converting sums like that...

Hi, I'm studying infinite series and am really struggling with memorizing all the tests for convergence in my book, there's like 10 of them. I don't think I'm going to be successful in memorizing all of them. I will never be asked in my course to use a specific test to determine convergence...

Homework Statement Hi, I'm trying to solve the problem in the attachment. I was asked to evaluate the left hand side equation of the equal sign. I was unsure how to go about evaluating it so I consulted my solutions manual to look up the first step. The right hand side equation of the...

On the provided attachment, I have problem understanding the last step of the 1st sum of that page. Can anyone explain to me what is being done after the second-last step?

Homework Statement I have to determine whether the series converges or diverges. \sum (cos^2 (n)) / n^2 +1 Homework Equations Suppose An and Bn are series with positive terms. If the limit of An over Bn as n approaches infinity equals C, and C is a finite number greater than 0...

Hi, guys. For an E&M/quantum mechanics problem I have to integrate the series below: Homework Statement Integrate \int{r^{2}e^{y r} \, dr}.2. The attempt at a solution Using integration by parts and and "differentiating under the integral" give the same answer: I= \frac{2...

Homework Statement What is wrong with the following calculation using infinite series? 0 = 0 + 0 + 0 + ... 0 = (1 - 1) + (1 - 1) + (1 - 1) + ... 0 = 1 - 1 + 1 - 1 + 1 - 1 + ... 0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... 0 = 1 + 0 + 0 + 0 + ... 0 = 1 Homework Equations None...

Homework Statement \sum_{1}^{inf} k^2/(n^3+k^2) The Attempt at a Solution I think it's Riemann but i cannot find a suitable function to integrate.

So tonight there was a new episode of Futurama. The professor created something that could create 2 copies of something. Bender got ahold of it, and started duplicating himself. Bender duplicated himself once, giving 3 benders. Then those 2 duplicates duplicated themselves, giving 7 benders...