Infinite series Definition and 390 Threads

Homework Statement Given the Infinite Series 1/(1+n^2) where n goes from 1 to infinity, show that the sum is less than pi/2. Homework Equations 1/(1+n^2)dx=arctanx Series goes 1/2, 1/5, 1/10, 1/17, 1/26 and so on The Attempt at a Solution I have tried to find a telescoping...

Homework Statement Test the series for convergence or divergence \sum^{\infty}_{1} \frac{(n!)^{n}}{n^{4n}} Homework Equations Ratio test seems likely since it contains n!, but I think I'm missing something The Attempt at a Solution :confused: After a failed attempt to use...

Given that the series from n = 1 to infinity of An converges to L, which of the following conclusions is valid for the series from n = 1 to infinity of (An)^2? A) It may diverge B) It converges absolutely C) It converges to M < L D) It converges to M > L E) It converges to M^2 = L My...

The series from n = 1 infinity of 1/(n*(3^n)) must A) converge to a value greater than 1/4 B) converge to a value greater than 1/9 C) Converge to a value less than 1/8 D) converge to a value less than 1/2 E) diverge. I know the series definitely does not diverge because the series...

Okay, there's two questions, actually. First, determine if the series converges. SUM: (n-2)/(n^2-4n) (from n=5 to infinity) I used the integral test, found the integral to be 1/2 log(n^2-4n) from x=5 to x=t as t approaches infinity. That turned out to go to infinity so the series...

Homework Statement Hi, How do i determine de result in terms of x of this series for x < 1: (Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i) Thanks The Attempt at a Solution I know that (Sum(i=0..+infinity;x^i) will tend do 1/(1-x) but i don't know what the numerator...

Homework Statement Does the infinite series \sum_{n=1}^{\infty}(\frac{1}{n^(1 + (\frac{1}{n}))}) converge? Homework Equations Power series \sum_{n=1}^{\infty}\frac{1}{n^p} The Attempt at a Solution I used the fact that for a power series, if p>1 the series will converge. Since...

Homework Statement \sum\frac{1}{n2^(n+1)} from 1 to infinity. By the way, that 2 is to the power of (n+1), doesn't show clearly. Homework Equations The Attempt at a Solution I have worked out the first few individual calculations, up to n=6, and i know it approaches ln(2)/2, however I...

Homework Statement Determine convergence for each of the following: ∞ ∑ 1 / [n (log n)2] n=2 ∞ ∑ 1 / [n log n log(log n)] n=2 [log=ln=natural log] Homework Equations The Attempt at a Solution I learned the root test, ratio test, comparsion test, and integral test. But...

Homework Statement Determine whether the following series converges absolutely, converges conditionally, or diverges. \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3 - ln(n)} Homework Equations The assortment of different tests. The Attempt at a Solution Okay, first of all, I tried using...

Homework Statement \sum_{n=1}^{\infty}\frac{sinn}{2^n} Homework Equations Definition of a geometric series: \sum_{n=0}^{\infty}x^n=\frac{1}{1-x} The Attempt at a Solution Basically I can use the geometric series idea and implement it into the denominator of the question (i.e. sub x=2 into...

Homework Statement Does it converge or diverge? Sum n=0 to infinity : (n/(n+1))^(n^2) The attempt at a solution I know I need to use the root test. But what I get is ... Limit to infinity : (n/(n+1))^n It seems that the n/(n+1) would go to 1 because when you multiply the top...

http://www4a.wolframalpha.com/Calculate/MSP/MSP167199e5bhg1gg5673i000048ed16i8cbf5iacg?MSPStoreType=image/gif&s=14&w=256&h=40 for all x in the interval of convergence of the given power series. a) write the first 3 nonzero terms and the general term for an infinite series that represents...

Homework Statement Well I am analysing the convergence of the following series: \sum\frac{2n}{n^{3}+1}x^{n} from n=0 to infinity. The attempt at a solution I have begun by using the ratio test, but as i have the limit in terms of x and n, i can't tell if it is bigger than 0? So does...

I am trying to understand how to calculate the sum of the following infinite series, can someone help please: (5/7)2 - (5/7)3 + (5/7)4 - (5/7)5 + ... The sum of such a series should be given by: a / (1-r) But the value of a = 0 (the first term = 0), hence my confusion. Thanks

Homework Statement Determine whether the series below is convergent or not: \sum 7*\frac{n!}{n^{n-10}} n=8 and the series goes to infinity (Sorry, I couldn't get the formatting correct.) Homework Equations n/aThe Attempt at a Solution Well, originally I thought the series was divergent...

Homework Statement I'm kind of new around here...Hope this forum will help me be a better future scientist :) I need some serious help in the attached questions...I need to determine whether the series in the picture converge, absolutely converge ot diverge... I really need your guidance...

Hi, sorry i wasnt quite sure where to post this. I think i know how to do it but have not encountered a question like it and don't have a mark scheme so thought id post it up to see if my thinking is correct. Homework Statement (c) By considering the integral of 1/(x^3) between N and...

Homework Statement I need help in the next questions. Prove or find counterexamples for the next propositions: 1. If the series [ Sigma (from n=1 to infinity) n*an ] converge then the series [ Sigma (from n=1 to infinity) n*a(n+1) ] also converge. 2. If [Sigma (n=1 to infinity) of an...

Find the sum of the infinite series \sum _{n=1}^{\infty } \left( i /2\right) ^{2\,n} I just can't seem to get started on this problem, so I was hoping somebody could give me a hint, as to what methods i should read up on.

Homework Statement Given that a_{n} > 0 and lim(na_{n}) = l with l\neq0, prove that \sum a_{n} diverges.Homework Equations The Attempt at a Solution lim(na_n)=l (with =/= 0), so I can safely say that: \left|na_{n}-l\right| < \epsilon by the definition of limit. Then isn't it also true that...

Homework Statement Prove whether \sum \frac{1}{ln(e^{n}+e^{-n})} converges or diverges Homework Equations The Attempt at a Solution (second post today... sorry, I just want to make sure I'm getting this right) Since e^{n}+e^{-n} goes to infinity as n goes to infinity...

Homework Statement \sum \frac{(-1)^{n}}{n+n^{2}} Does this series converge as n -> infinity? Homework Equations The Attempt at a Solution First, by the absolute convergence test, \sum \frac{(-1)^{n}}{n+n^{2}} should converge if \sum \left|\frac{(-1)^{n}}{n+n^{2}}\right|...

Homework Statement \sum^{\infty}_{n=1} \frac{n}{2^{n}} Does this series converge or diverge? Homework Equations The Attempt at a Solution By the Cauchy condensation test (http://en.wikipedia.org/wiki/Cauchy_condensation_test) I think this one diverges. But not sure if I am...

Homework Statement \sum^{\infty}_{n=1} \frac{(-1)^{n}}{\sqrt{n}+(-1)^{n}} Prove whether this series converges or diverges using the following analysis: Let b_{k} be the sum of terms numbered n=2k-1 and n=2k from the given series. By simplifying b_{k}, determine if \sum^{\infty}_{k=1}...

Homework Statement Consider the infinite series ∞ ∑ (-1)n+1 cos(nx) n=1 Is there any convergence? (Answer: No convergence) 2. Homework Equations /concepts Infinite series The Attempt at a Solution I was looking back in my 1st year calculus textbook, but none of the theorems seem...

Homework Statement The capacitance of each capacitor of the infinite series shown in the picture is C = 1\muF. Find the total capacitance between points a and b. IMAGE: http://img61.imageshack.us/img61/3674/pic002311.jpg" (continues to infinity) Homework Equations In series, (1/Ceq) =...

Homework Statement Find the limit of the sum of: y = (2n + 3n) / 4nThe Attempt at a Solution as n-> infinity, y approaches 0. I don't know where to proceed from here.

I can't remember much from my intro. analysis class anymore. If you have an infinite series that ultimately converges, can the first few terms diverge (i.e., can they move away from the convergence point)? And if so, how many of these terms can do so? I'm trying to understand how to "get...

Homework Statement Using the comparison test determine if the infinite series for sin(3/n^2) converges or diverges. The Attempt at a Solution Well... these are pretty straight forward, and it's pretty obvious that this is convergent, but I'm having trouble applying the...

I have to solve the nonlinear DE y'=x²-y² by using an infinite series expansion y=\sum_{n=0}^{\infty} a_n x^n, but I've tried in vain. Maybe a change of variables would make it easier, but I don't know which one. Thanks

a couple of "Infinite Series" questions... Hi, I'm trying to solve some problems in "Stroud's Engg mathematics"... I'm stuck with these 2 questions: Σ (r=0→∞) (2r) / (r+1)! and Σ (r=0→n) (2r-1) / r(r+1)(r+2) the 1st question converges to 2. My 1st try is divide everything the...

I read a proof for showing that the harmonic series is a diverging one. This particular one used a comparison test: 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... + 1/16 + ... 1/2 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 +... + 1/16 + ... Each term in the second series is <...

I just need to make sure that I've got this analysis right: The argument S = 1 - 1 + 1 - 1 + 1 - ... then S = (1 - 1) + (1 -1) + (1 -1) + ... = 0 is invalid because it ignores all sum Sn for n not congruent modulo 2 (not even). The argument S = 1 - 1 + 1 - 1 + 1 - ... then S = 1 - (1 -1) - (1...

Hi, I wonder if this hypothesis is true: Let f_n be an arbitrarily chosen n'th anti-derivative of the function f_0. Similarly, let g_n be the n'th derivative of the function g_0. Now, \int^b_a f_0 g_0 \rm{d}x=[f_1g_0]^b_a-\int^b_a f_1g_1 \rm{d}x=[f_1g_0-f_2g_1+...]^b_a+(-1)^n \int^b_a...

\sum^{\infty}_{x=1} \frac{cos(14.1347 \ln (x))}{x^{a}} = 0 Is there a way to solve for a? I don't think so but maybe someone here will have an insight as to what to do..

\Sigma_{n=1}^{ \infty} \frac{1}{(3n-2)(3n+1)} I simplified it to partial fractions to (1/3) / (3n-2) - (1/3) / (3n+1) Now what?

Homework Statement Show that for all integers n \geq 1, cos(2x) + cos(4x) + ... + cos(2nx) = \frac{1}{2} (\frac{sin((2n+1)x)}{sin(x)}-1) Use this to verify that \sum_{n=1}^{\infty}(\int_{0}^{\pi} x(\pi-x)cos(2nx)dx) = \frac{-1}{2}\int_{0}^{\pi} x(\pi-x)dx) Homework Equations...

Homework Statement Let T(x) = \sum^{\infty}_{k=0} \frac{1}{2^k} \frac{(x-3)}{k!}k be the Taylor series for a function f. What is the value of f10(3), the tenth derivative of f at x = 3? The Attempt at a Solution I have a very small idea of actually starting this problem. Can I just...

Homework Statement I know that the ratio test can be proved using the geometric series knowledge, but I'm looking for a way to prove the ratio test without the geometric series at all. The reason is that I'm proving the geometric series convergence with the ratio test, and my professor...

Homework Statement \sum\frac{7^{k}}{5^{k}+6^{k}} Determine if this infinite series (from k=0 to infinity) converges or diverges. 2. The attempt at a solution I set ak=\frac{7^{k}}{5^{k}+6^{k}} then I took the Ln of both sides ln ak=ln\frac{7^{k}}{5^{k}+6^{k}}=ln7k-ln(5k+6k) I'm not...

Homework Statement Let f(x) be a function with the following properties. i. f(0) = 1 ii. For all integers n \geq 0, the 0, the nth derivative, f (n)(x) = (-1)nanf(x), where a > 0 and a \neq 1. a.) Write the first four non-zero terms of the power series of f(x) centered at zero, in...

Homework Statement Let f be a function whose seventh derivative is f7(x) = 10,000cos x. If x = 1 is in the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at x = 0 will approximate f(1) with an error of not more than a.)...

Homework Statement Determine the sum of the following series: \sum_{n=1}^{inf} log(1-1/(n+1)^2) Sorry for poor latex, that is supposed to say infinity. Homework Equations How might we turn this into an easier function to deal with? The Attempt at a Solution So far I've only...

Homework Statement Find the sum \sum_{0}^{\infty} \frac {n^2} {3^n} Homework Equations The Attempt at a Solution I don't know how to go about finding this sum, I have a guess of what it will be just by adding the first ten terms or so, but how do I find an actual approximation?

Infinite Series without an "x" term What are the uses of an infinite series which does not include an "x" variable? If you are looking at an infinite series which sums terms based entirely upon each term's position in the series, or, a series which includes only the variable "n," how would...

My question is one of vocabulary. What does it mean to evaluate an infinite series in closed form? If I have a Series: \Sigma 1/ (N2), as N goes from 1 to infinity. This is similar to a test question I'm working on so, I DO NOT want to know how to solve it, I just want to know exactly...

Homework Statement \sum^{infinity}_{n=1} (1/n^2 - 1/n^3) Homework Equations it goes to infinity n=1 The Attempt at a Solution Im assuming this is a telescoping series. when I plugged in my terms nothing canceled out except for the 1's at the beginning. (1-1)+(1/4 - 1/8)+(1/9 -...

I need some help on determining whether this infinite series converges (taken from Spivak for those curious): \sum_{n=1}^{\infty } \frac{1}{n^{1+1/n}} I would think the integral test would be most appropriate but it doesn't seem to work (because the integral seems hard). The obvious comparison...

Homework Statement so I have learned how to do different problems of series. but there's this problem that I spent hours last night but could not come up with anything. which is- ∞ ∑ n/2^(n-1) n=1 Homework Equations I have no idea The Attempt at a Solution so from there...