Infinite series Definition and 386 Threads

Given, U(n)=1/(logn)^(2*n) To find: Whether the series ƩU(n) is convergent or divergent. Sequence of tests to be followed: *Comparison tests *Integral tests *D'Alembert's ratio test *Raabe's test *Logarithmic test *Cauchy's root test My approach: Comparison test: Since the series V(n) cannot...

The given series is: 1+[(a+1)/(b+1)]+[(a+1)(2*a+1)/(b+1)(2*b+1)]+[(a+1)(2*a+1)(3*a+1)/(b+1)(2*b+1)(3*b+1)]+...∞ Problem: To find U(n+1)/U(n). My approach: Removing the first term(1) of the series and making the second term the first,third term the second and so on... I get...

On the bottom of page 24 & top of page 25 of this pdf an integral is beautifully computed by breaking it up into an infinite series. Is there any reference where I could get practice in working integrals like these?

Homework Statement Determine convergence or divergence using any method covered so far*: Ʃ(1/(n*ln(n)^2 - n)) from n = 1 to infinity*The methods are the following: - Dichotomy for positive series (if the partial sums are bounded above and the series is positive, the series converges) -...

Homework Statement Determine whether the series diverges or converges. (1+2) / (1+3)+ ((1+2+4)/(1+3+9))+ ((1+2+4+8)/(1+3+9+27)) + ...The Attempt at a Solution I have split up the series into two (denominator and numerator): an = (1+2) + (1+2+4) + (1+2+4+8)+... = (1)n + 2n + 4(n-1) + ... bn...

Trying to solve this infinite series?? Hey folks! I've spent hours trying to solve this and have exhausted all available resources.. I just need to be pointed in the right direction! Homework Statement Compute the sum of the infinite series (I believe this is an arithmetico geometric...

Homework Statement Preliminary test: If the terms of an infinite series do not tend to zero, the series diverges. In other words if ##\lim_{ n \to \infty}a_n \neq 0## then the series diverges. But if the limit is 0 we have to test further. Suppose a series a series satisfy this condition...

Find $$\sum_{n=0}^\infty \frac{\cos(nx)}{2^n}$$.

Homework Statement The sum of the infinite terms of the series \text{arccot}\left(1^2+\frac{3}{4}\right)+\text{arccot}\left(2^2+\frac{3}{4}\right)+\text{arccot}\left(3^2+\frac{3}{4}\right)+... is equal to A)arctan(1) B)arctan(2) C)arctan(3) D)arctan(4) Ans: B Homework Equations The Attempt at...

hello ... I propose this exercise for you to solve on various methods ...\[\lim_{n \to{+}\infty}{\frac{1}{n}\sum_{i=1}^n({1+\frac{i}{n}}})^{-2}\]thanks att jefferson alexander vitola(Smile)

How to find the value of an infinite series. for e.g.Ʃ_{n=1}^{\infty} (β^{n-1}y^{R^{n}}e^{A(1-R^{2n})}) where β<1, R<1, y>1, and A>0? Note that this series is covergent by Ratio test. I already have the numerical solution of the above. However, I am interested in analytical solution...

I was looking at this topic: http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys-closed And the top answer uses the fact that the sum from 1 to infinity of 1/(x2^x) is log 2. Why is this true? Thanks in advance.

Given S, an Infinite Series Summation, find \frac{1728}{485}S S=1^2+\frac{3^2}{5^2}+\frac{5^2}{5^4}+\frac{7^2}{5^6}+... I found out the formula for (r+1)th term of the series, hence making the series asS=1+\sum_{r=1}^{\infty}\frac{(2r+1)^2}{(5^r)^2} Now I have a hard time guessing what to do...

Find the series sum ln2/2 – ln3/3 + ln4/4 – ln5/5 + ….

Hi everyone ;) I have a challenging problem which I would like to share with you. Prove that \[\frac{1}{2^2}+ \frac{1}{3^2} \left(1+\frac{1}{2} \right)^2+\frac{1}{4^2} \left( 1+\frac{1}{2} +\frac{1}{3}\right)^2 + \frac{1}{5^2} \left( 1+\frac{1}{2} +\frac{1}{3}+\frac{1}{4}\right)^2 +\cdots=...

Homework Statement I need to fin the sum of the following two infinite series: 1. Ʃ[n=0 to ∞] ((2^n + 3^n)/6^n) and 2. Ʃ[n=2 to ∞] (2^n + (3^n / n^2)) (1/3^n) Homework Equations use the sum Ʃ[n=2 to ∞] (1/n^2) = ∏^2 / 6 as necessary The Attempt at a Solution I tried to manipulate them...

Homework Statement I need to use the Comparison Test or the Limit Comparison Test to determine whether or not this series converges: ∑ sin(1/n^2) from 1 to ∞ Homework Equations Limit Comparison Test: Let {An} and {Bn} be positive sequences. Assume the following limit exists: L =...

Test these for convergence. 5. infinity E...((n!)^2((2n)!)^2)/((n^2 + 2n)!(n + 1)!) n = 0 6. infinity E...(1 - e ^ -((n^2 + 3n))/n)/(n^2) n = 3 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?

Test these for convergence. 3. infinity E...((-1)^n)*(n^3 + 3n)/((n^2) + 7n) n = 2 4. infinity E...ln(n^3)/n^2 n = 2 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?

Test these for convergence. 1. infinity E...n!/(n! + 3^n) n = 0 2. infinity E...(n - (1/n))^-n n = 1 Btw, E means sum. Which tests should I use to solve these?

Homework Statement Evaluate the indefinite integral as an infinite series ∫ sin(x2) dx Homework Equations The Macluarin series of sin x = ∞ Ʃ (-1)nx2n+1/(2n+1)! n=0 The Macluarin series for sin(x2) = ∞ Ʃ (-1)x4n+2/(2n+1)! n=0 The Attempt...

For the series such that: \Sigma _{n=1} ^{\infty} a_n =\Sigma _{n=1} ^{\infty} b_n A certain theorem says that these series are equal even if a_n = b_n only for n>m. That is, even if two infinite series differ for a finite number of terms, it will still converge for the same sum. I am thinking...

Homework Statement ∞ Ʃ (-1)^(k+1) / kln(k) k=2 Homework Equations integral test, p test, comparison test, limit comparison, ratio test, root test. The Attempt at a Solution In class so far we have not learned the alternating series test so i can't use that test. So far I have...

If $\displaystyle y=\frac{3}{4}+\frac{3*5}{4*8}+\frac{3*5*7}{4*8*12}+...\infty$. Then $y^2+2y = $

How to make a surface plot involving an infinite series in Matlab Solving Laplace's equation for electric potential for a 2D surface yields: V(x,y) = 4 Vo/pi * Ʃ (n=1,3,5,...) (1/n e^(-npi*x/a) sin(n*pi*y/a) ,where a is and Vo are constants ...it's convoluted, but basically, I need to...

Homework Statement Find the radius of convergence and interval of convergence for the following infinite series \sum_{n=1}^{\∞} \frac{x^n n^2}{3 \cdot 6 \cdot 9 \cdot ... (3n)} Homework Equations Ratio test The Attempt at a Solution Using ratio test we get im not sure how to...

Homework Statement How many terms of the series do we need to add in order to find the sum to the indicated accuracy? Ʃ((-1)n)/(n(10n)) from n=1 to infinity |error| <.0001 I keep ending up with n=log(4)-log(n)

Homework Statement Show that the infinite series \sum_{n=0}^{\infty} (\sqrt{n^a+1}-\sqrt{n^a}) Converges for a>2 and diverges for x =2The Attempt at a SolutionI'm reviewing series, which I studied a certain time ago and picking some questions at random, I can't solve this one. I tried every...

Hi everybody, while doing a complex analysis exercise, i came to a strange inequality which i don't know how to interpretate. Suppose you have a sequence $\{a_j\}$ of positive real number. Let $\rho$ a positive real number. The inequality i found after some calculation is...

Homework Statement cot^-1 3 + cot^-1 7 + cot^-1 13+... Homework Equations The Attempt at a Solution I first tried to write the nth term of the series t_n = cot^{-1}\left( 2^n + (2n-1) \right) Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I...

1. Homework Statement ∑ i=1 to n1+(1/i2)+(1/(1+i)2)−−−−−−−−−−−−−−−−−−−−√ = n(n+2)/n+1 2. The attempt at a solution First I did the base case of p(1) showing 3/2 on the LHS equals the 3/2 on the RHS. Then I assumed p(k) and wrote out the formula with k in it. Then prove p(k+1)= p(k)+...

Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...

$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.

Homework Statement The problem is part of a review and we are only to determine if the series converges or diverges by any test, and state the test. ##\sum_{n=1}^\infty(\frac{k}{k+1})^k## My work so far I know that the root test gives an inconclusive answer and from there I moved...

Question says: \sum(cos(n*pi)/5^n) from 0 to infinity. Proved that it converges: ratio test goes to abs(cos(pi*(n+1))/5cos(pi*n)) with some basic algebra. As n goes to infinity, this approaches -1/5 (absolute value giving 1/5) since cos(pi*(n+1))/cos(pi*n) is always -1, excepting the...

I am having a difficult time seeing how \sum_{n=0}^{\infty} ((-1)^n + 1)x^n is equivalent to 2\sum_{n=0}^{\infty} x^{2n}

Homework Statement I need to show that both sin(x) and cos(x) are absolutely convergent. Here's my work so far, Theorem: ℯix = cos(x) + i*sin(x) (1) Proof: This...

This might sound like a dumb question, but it's actually not too obvious to me. If we know that \lim_{n→∞}S_{n} = L , can we prove that \lim_{n→∞}S_{n-1} = L ? I'm actually using this as a lemma in one of my other proofs (the proof that the nth term of a convergent sum approaches 0), but...

S = \frac{1}{2} + \frac{1}{4} + ... + (\frac{1}{2^n}) I noticed that this is a sum of a infinite series with the common ratio being 1/2, so using \frac{1}{1-1/2} I get S = 2, however with this question there is a hint saying multiply S by 2, which I did not use so I'm worrying if I done...

Homework Statement I attached the solution to the problem. Homework Equations The Attempt at a Solution I can see that the infinite series diverges by looking at a few terms, but how would I find a general term for the infinite series, to evaluate it analytically?

Hi, In griffith's "Introduction to Electrodynamics" he indicates that a specific infinite series has a truncated form (the series and truncated form are given below) And he says the reader can try to show that it indeed has that form...

Recently I have been playing around with infinite series and related topics, when I realized I couldn't figure out how to solve something of the form \sum_{n=1}^{\infty}\frac{n}{k^{n}} How would you go about finding a sum like this?

Hello. Having already learned about infinite series and sequences in my calculus class, I'm quite interested in them and especially in learning more about them. If any of you have in mind any good books on the subject which you can recommend to me, it will be very much appreciated...

Homework Statement 1) \displaystyle \sum\limits_{n=1}^{\infty }{\frac{{{n}^{3}}}{{{e}^{n}}}} 2) \displaystyle \sum\limits_{n=1}^{\infty }{\frac{-1}{2\left( n+1 \right)}+\frac{2}{\left( n+2 \right)}-\frac{3}{n\left( n+3 \right)}}Homework Equations The Attempt at a Solution I've proved that...

Homework Statement Does \sum _{ n=1 }^{ \infty }{ \frac { { \alpha }^{ n }{ n }! }{ { n }^{ n } } } converge \forall |\alpha |<e and if so, how can I prove it? Homework Equations { e }^{ x }=\sum _{ n=0 }^{ \infty }{ \frac { { x }^{ n } }{ n! } } The Attempt at a Solution...

Homework Statement This isn't my homework problem, but I'd like to know how to prove \sum _{ n=0 }^{ \infty }{ \frac { n }{ { \alpha }^{ n } } } =\quad \frac { \alpha }{ { (\alpha -1) }^{ 2 } } I only know basic convergence tests (including the integral test), and that \forall |\alpha...

Homework Statement I need to find the sum of a given infinite series when |x|<1 (which is the radius of this series) Homework Equations [SIZE="3"]\sum_{n=1}^{∞}(-1)^{n+1}\frac{x^{2n+1}}{4n^2-1} The Attempt at a Solution I've tried to do the following: [SIZE="3"]S'(x) =...

What is the LaTex and infinite sum for 1-2-1/2+3-1/2-4-1/2+5-1/2 . . . Does it converge anyway? I am too old for this to be a school assignment.

Homework Statement Find the equivalent impedance of the infinite series of resistors and capacitors as shown below [FONT="Courier New"] -R----R----R----R----...R----... ____C____C____C____B_______C -r----r----r----r----...r----... Homework Equations 2.1. Equivalent resistance of resistors in...

Homework Statement Find the sum of infinite the series (-1)^n * 7^n/n! for n=1 to infinity Homework Equations e^x = sum (x^n)/n! for n=0 to infinity The Attempt at a Solution I combined the (-1)^n and the 7^n to make the summation ((-7)^n)/n! for n = 1 to infinity then I changed the lower...