infinite

What are the values of k that make the series convergent?

Find the indicated limit. lim as x goes infinity of |(x + 1) / (x - 2)|^(x^2 - 4)^(1/2)

Lim -x / (4 + x^2)^(1/2) as x goes ∞ = -1 I don't Understand why the answer is -1 because when I look at the graph of the function I see when x goes ∞-, y goes 1 but when x goes ∞+, y goes -1 since both are not equal, it must be DNE. Where is the problem of the graph?

Determine the value of constant c such that \sum _{ n=2 }^{ \infty }{ { (1+c) }^{ -n } } = 2 Apologies if i'm missing something really obvious but should this be solved simply by observing that the denominator of each term becomes infinitely smaller and thus solving the polynomial?

How would I go about computing the square of a series of the form: \Sigma_{j=0}^{\infty}[B_jsin(\Omega_j)+C_jcos(\Omega_j)] I imagine this would involve a Cauchy product, but I'm not sure how to get started.

Hello, I would like to use the Girsanov transformation in the Black-Scholes model with infinite time horizon in order to find an equivalent martingale measure. The problem is not the Girsanov transformation itself but the existence of the probability measure. Here is a more detailed...

I've tried to find the sum of this infinite series infinity sum = 1/(k(k+3)) k=1 I tried to treat it as a telescoping series.I decomposed the fractions and I've found a pattern where the negative part of a_2 cancels with the positive part of a_6.I then figured that...

Hi I have two questions asking me to find the interval of convergence of an infinite series: 1. Sum n=1 - infinity of: (2x)^n/(n^2+1); and 2. The Taylor series (or actually McLaurin series I think) for ln(1+x) about x=0 ie sum k=1 to infinity ((-1)^k*x^k)/k The actual words I have used are...

(sorry, I don't know in which section I should post this question) I now think I have some idea why Cantor (or whoever it is) said things like "there are more real numbers R than whole numbers N." So I think I've understood the concept of comparing infinite sets. And why this is...

Hello, New to this topic after missing it at uni due to sickness! Would anyone be able to help me to do this question as im not really sure how to prove that something is convergent for all values of X. [I do understand how to show that a series converges for a certain value of X]

$A_n=\{n,n+1,n+2,...\}$ Can't see how it follows from the last sentence. Can someone explain?

The problem is: $\sum_{n=1}^\infty nx\ (\ \prod_{k=1}^n \dfrac{sin^2(k\theta)}{(1+x^2+cos^2(k\theta))}\ )$ The way that I tried solving it is using ln. $ln(S_n)=ln(\ (\sum_{n=1}^\infty nx)\ (\ \prod_{k=1}^n \dfrac{sin^2(k\theta)}{(1+x^2+cos^2(k\theta))})\ )$ turns to... $ln(S_n)=ln(\...

I'm back. Managed to make it from chapter 3 to chapter 8, almost at chapter 9, with a virtually difficulty-free journey - but before I can take that final leap on to Series, I must become un-flummoxed. Larson's Calculus ET; chapter 8 section 8: Improper Integrals It briefly mentioned in the text...

Please tell me what I am doing wrong.

Q: A ball is dropped from a height of 32 ft. Each time it strikes the ground it rebounds 3/8ths of the distance from which it had fallen. Theoretically, how far will the ball travel before coming to a rest? Because the ball rebounds up then falls back down again before rebounding a next time, I...

By integrating 1/x, 1<x<k we get the volume of V=pi(1-1/k) units^3, limit V k--> infinity is pi units ^3. doing the exact same for 1/x^1/2 we get V= pi ln(k) units^3, and as this approaches infinity we get an infinite volume. This is all to do with Gabriel's horn. This shape is of course created...

Imgur: The most awesome images on the Internet I have no idea on this one, so could someone explain it or lead me to resources that would help me solve it? Imgur: The most awesome images on the Internet For this one, could someone help me on C, and check my answers for the others? I got...

Imgur I know how to prove that each is convergent, but I'm not sure how to use the 1/4^n infinite series to do it. http://imgur.com/mbdfyrL,MsOGzLV,KI34PbG,gnAF0DK,u0L9kV6#2 Also, for this one, I'm not sure how to set up the limit. Could someone help me, or direct me to resources that...

Hi all, an interesting infinite series question for you: show that $\displaystyle \sum_{r=1}^\infty 2^{1-r} \tan\left(\frac {x}{2^r} \right ) = \frac 2x - 2\cot x.$ It was posed to me by someone else and I don't much care for the actual solution, but any tips would be welcome. Thanks!

So, on a quiz, I used the direct comparison test instead of using the limit comparison test and my teacher marked me wrong.... I'm not sure why using the direct comparison test in the following test is invalid: \sum_(n=5)^\infty \frac{\sqrt{n+1}}{\sqrt{n^4+2n^2+3}} using the limit comparison...