I am really stuck on this problem. I don't really understand power series expansions. However, I think this has to do with generating functions.
124038
What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$?    How do I do power series expansions?
4742
I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms):  $f = (1 + \epsilon \,x)^{1/\epsilon}$, where $\epsilon$ is a small parameter. I'm sure this is very simple, but I wasn't certain about the best way to approach this. A quick look at mathematica tells me the solution is $e^x - \frac{1}{2} (e^x x^2) \,\epsilon + ...$. How would I go about getting this answer - and more importantly, how would I systematically find series expansions for problems similar to this one?
31037
An  which I am not able to solve.  Let $\mathsf{S_{n}}$ denote the $n$-th partial sum of the series: $$\sum\limits_{r=1}^{\infty} \frac{1}{r(r+1)}$$  Prove that for every integer $k&gt;1$, there exists integers $m$ and $n$, such that $$s_{m}-s_{n} =\frac{1}{k}$$
121291
I have a clueless friend who believes that   $$ \sum_{n=1}^\infty \frac{1}{2^n} $$  doesn't equal $1$ in the 'normal arithmetical sense'.   He doesn't believe that this series flat out equals $$1$$ Is he correct?
120508
Given $a_0 = 1$ and $a_n = (n + 1)a_{n - 1} + 3^n$ for $n \geq 1$, find formula for $a_n$.   First remind the definition of power series $F(x) = \sum_{n \geq 0}f_n \frac{x^n}{n!}$ and exponential generating function $G(x) = \sum_{n \geq 0}\frac{x^n}{n!} = e^x$   Note we have to use exponential generating function to solve this question. So let $f(x) = \sum_{n \geq 0}a_n\frac{x^n}{n!}$, multiplying both sides of  $a_n = (n + 1)a_{n - 1} + 3^n$ by $\frac{x^n}{n!}$ and sum over all $n \geq 1$. We get, $$\sum_{n \geq 1}a_n\frac{x^n}{n!} = \sum_{n \geq 1}(n + 1)a_{n - 1}\frac{x^n}{n!} + \sum_{n \geq 1}3^n\frac{x^n}{n!}$$ $$\sum_{n \geq 1}a_n\frac{x^n}{n!} = \sum_{n \geq 1}na_{n - 1}\frac{x^n}{n!} + \sum_{n \geq 1}a_{n - 1}\frac{x^n}{n!} + \sum_{n \geq 0}\frac{(3x)^{n}}{n!} - 1$$ $$\underbrace{\sum_{n \geq 0}a_n\frac{x^n}{n!}}_{f(x)} - 1 = x\underbrace{\sum_{n \geq 1}a_{n - 1}\frac{x^{n-1}}{(n-1)!}}_{f(x)} + \underbrace{\sum_{n \geq 1}a_{n - 1}\frac{x^{n}}{(n)!}}_{\int{f(x)} = F(x) + C} + e^{3x} - 1$$ $$F'(x) - 1 = xF'(x) + F(x) + e^{3x} - 1   + C$$ $$F'(x)(1 - x) - F(x) = e^{3x} + C$$ I think I'm essentially stuck here, I'm trying my best to manipulate each term so that we can express each term in terms of $f(x)$. But I can't go further from here because it doesn't like look this form can lead us to the solution.   Any feedback or suggestion?
67183
Consider the following complex power series :$$\sum_{n=1}^\infty\frac{z^n}{n}$$ The radius of convergence of this series is $1$ and the series is divergent for $z=1$. I want to know what are the values of $z\in C:=\lbrace z\in\mathbb{C}: |z|=1\rbrace$, the circle of convergence, for which the given series converges.
73531
If you expand $(x_1+x_2+\cdots+x_k)^n$, how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients?  I'm kind of lost here. This came up with other questions on the topic of binomial coefficients, and i have figured those out. I have spent almost an hour trying to figure where to start and haven't made any progress.
108215
Studying for an exit exam and it's been years since I work with any series expansions. Here's a past problem:  Show that $$\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$$ converges to an analytic function on the set $A=\{z \in \mathbb{C}: |z|&lt;1 \}$.  I'm looking to use Taylor's theorem for this problem but how would I compute the radius of convergence? Do I have to do a comparison test?
58107
$u$ substitution doesnt work. I don't see any connection with the Weierstrass substitution either. integration by parts results in a infinite integral series.
84469
Find the value of:  $i^{i^{i^{i^{i^{i^{....\infty}}}}}}$  Simply infinite powering by i's and the limiting value.  Thank you for the help.
87518
How do I find the sum of this series?   $$\sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}$$   Hints in the right direction would be appreciated.
60872
I've tried solving it, but I'm not familiar with these kind of progressions and I couldn't get to a result. Also, I've found that the result is just $N/2$, but I don't get why would that be the answer, as it doesn't work for $N=1$ or $N=2$. Appreciate your help :)
25938
$\dfrac{n}{2} \le \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n - 1} \le n $  I've Tried for hours but didn't got any striking idea. I don't have any efforts to show rather than induction. Please Try If you 've non-inductive proof.
10854
I am trying to prove this inequality  $$|e^z-1|\leq e^{|z|}-1\leq |z|e^{|z|}$$  I've tried calculating the difference in their power series   $$\sum_{k=0}^\infty\frac{|z|^k}{k!}-1-\left|\,\sum_{k=0}^\infty\frac{z^k}{k!}-1\,\right|$$   but did not get anywhere.
16750
I need help with how to compute the radius of convergence of the power series $\sum_{n=0}^{\infty} n!x^n$?  I was thinking of using the ratio test but am unsure of how to go about it. Any help would be greatly appreciated, thanks
50061
I have a summation series of the form: $n + n/2 + n/4 + n/8 + n/16 +\ldots  + 1$. Is there a formula for this series? Basically, the denominators are powers of 2.
33100
$1 + \frac{1}{3}\frac{1}{4} + \frac{1}{5}\frac{1}{4^2} + \frac{1}{7}\frac{1}{4^3} + ......$ Can anyone help me out how to solve this.   My try : I was thinking about the expansion of $tan^{-1}x$. But in that series  positive and negative will come alternatively.