Tagged Questions
2
votes
3answers
120 views
What is the answer to this limit
what is the limit value of the power series:
$$ \lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}}$$
where $m>1$.
5
votes
2answers
175 views
The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$
What is the answer to the following limit of a power series?
$$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
2
votes
1answer
42 views
Total area of squares.
We have a square whose length is $1$ unit. Every time we rotate by $\theta$ and scale the square such as you see in the image. Does the total area of squares converge if $\theta $ goes to $0$?
2
votes
2answers
51 views
Solving limit by substituting a power series
I dont understand why I am getting 2 and the textbook says it is -2.
$$\lim_{x\to 0} \frac{1-e^x}{\sqrt{1+x}-1}$$
I subbed the power series for $e^x$ and $(1+x)^{1/2}$ then got rid of the $1$ on top ...
5
votes
2answers
94 views
$\lim_{x\rightarrow\infty}\frac{f(x)}{e^x}$ for analytic functions
For some analytic function $f(x)=\lim_{n\rightarrow\infty}\sum^n_{r=0}c_rx^r$,
...
10
votes
1answer
174 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
3
votes
2answers
134 views
Is $\lim_{n \to \infty} \frac{n}{(n!)^\frac{1}{n}} = e$ any easier than Stirling? [duplicate]
Possible Duplicate:
Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
Stirling's approximation says that
$$ \lim_{n \to \infty} \frac{n^n \sqrt{n}}{n! e^n } = ...
12
votes
2answers
358 views
$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
Does the following limit exist? What is the value of it if it exists?
$$\lim\limits_{x\to\infty}f(x)^{1/x}$$
where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$ and $\{a_k\}\subset\mathbb{N}$ ...
2
votes
2answers
112 views
Approach to limit of infinite product
I was wondering if there is any proof that the limit of infinite product
$$\lim_{n \to \infty} \prod_{i=1}^{n} x_i, \mathrm{where}$$
$$0 < x_i < 1$$
is equal to 0 and that it does not ...
1
vote
1answer
480 views
list of convergent series
I wanted to know if there is an online reference I can use to find out known results about convergent series. I could not find this one, for example, on wikipedia
$\sum_{k=1}^{+\infty} ...
1
vote
2answers
338 views
Radius of convergence of power series $\sum_{n=0}^{\infty} \frac{(2x-5)^n}{n^2}$
Consider the below example:
$$\sum_{n=0}^{\infty}\frac{(2x-5)^n}{n^2},\qquad c_n=\frac{2^n}{n^2},\qquad R=2^{-1}=\frac{1}{2}.$$
$$\begin{align*}
\lim_{n\to\infty}\frac{c_{n+1}}{c_n} &= ...
