All Questions
12
questions
4
votes
1
answer
88
views
A conjecture involving series with zeta function
Recently, I tried to evaluate a limit proposed by MSE user Black Emperor. In the process of evaluating the limit, I have obtained the following equality.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 1,240
1
vote
0
answers
29
views
Determine the specific value of the division of two factorial series
I want to find a specific range of $\alpha$ formula as follows.
$$\alpha =\frac{1+\frac{{{x}^{2}}}{3!}+\frac{{{x}^{4}}}{5!}++\cdots+{\frac {x^{2n}}{(2n+1)!}}}{1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{...
- 285
0
votes
2
answers
169
views
Radius of convergence for $\sum_{n=0}^\infty n^nx^n$ and $\sum_{n=0}^\infty \frac {(-3)^n}{n}(x+1)^n$
How can one calculate the radius of convergence for the following power series:
$$\sum_{n=0}^\infty n^nx^n$$
and
$$\sum_{n=0}^\infty \frac {(-3)^n}{n}(x+1)^n$$
Regarding the first one I know ...
- 931
1
vote
3
answers
117
views
Suppose $\sum_{n=1}^{\infty}a_nx^n=f(x)$, then what can we say $\sum_{n=1}^{\infty}\frac{a_n}{n^3}x^n$ and the limit?
Let $f(x)=\frac{1}{(1-x)(1-x^4)}$, let $a_n$ be the nth term of the maclaurin expansion of $f(x)$.
What can we say about the power series $a_0+\sum_{n=1}^{\infty}\frac{a_n}{n^3}x^n$? Can we express ...
- 293
0
votes
1
answer
1k
views
Summation of infinite exponential series
How is the given summation containing exponential function
$\sum_{a=0}^{\infty} \frac{a+2} {2(a+1)} X \frac{(a+1){(\lambda X)}^a e^{-\lambda X}}{a! (1+\lambda X)}=\frac{X}{2} (1+ \frac{1}{1+\lambda X})...
- 93
1
vote
1
answer
62
views
Limit value using serie definition
I need to find the limit value of $$\lim_{x\to\:0}\frac{sin(\frac{1}x)}{\frac{1}x}$$
I wanted to do it with the serie definition of sinus and I come to the result:
$$ 1 -\lim_{x\to\:0} \sum_{i=0}^{\...
- 119
2
votes
1
answer
428
views
Limit of a sum of powers [duplicate]
I need to find the limit of a sequence indexed by $n\in \Bbb N$. $k$ is fixed natural constant.
The sequence is:
$x_n = \frac{1^k+2^k+3^k+\ldots+n^k}{n^k} - \frac{n}{k+1}$
I tried to solve this ...
- 131
5
votes
2
answers
495
views
How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?
How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?
So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$
Let $u=\arctan{(x)}$
$\lim\...
- 2,758
1
vote
2
answers
97
views
Limit of power series with L'Hospital
Calculate the given limit: $$\lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty\ n^5x^n$$
First, I used Taylor Expansion (near $x=0$): $$1-\cos(x^2)\approx 0.5x^4$$
I'm now quite stuck with the sum....
- 165
-1
votes
4
answers
66
views
Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?
Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?
I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has $\frac{t^{k+1}}{(k+1)!}$...
- 12.1k
0
votes
2
answers
153
views
Intervel of Convergence of a Power Series
Can anyone explain how to do this problem? I think you might be able to approach it with the ration test but I'm unsure. Any help is greatly appreciated!
$$\sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ \...
- 257
6
votes
1
answer
164
views
Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$
Simplify the following expression
$$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$
where $a$ is a real number and $f(x)$ is an analytic real function.
What is $\lim_n ...
- 1,258