All Questions
Tagged with sequences-and-series power-series
2,718
questions
7
votes
4
answers
187
views
Power series where the number $e$ is a root
I have been going at this question for weeks now and couldn't find anything.
Can we have a series of the form:
$$f(x)=\sum_{n=0}^{\infty} a_n x^n$$
where $a_n$ are rationals and not all $0$ such that $...
- 89
0
votes
0
answers
13
views
Convergence rate of Laguerre coefficients for polynomially bounded functions
Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies:
$$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$
for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for ...
- 685
0
votes
0
answers
30
views
Polynomial elevated to a decimal [closed]
I am a programmer who is trying to create a calculator in $\tt C\!+\!+$:
This calculator may use polynomials.
...
- 17
1
vote
0
answers
63
views
How to find the roots of sin(x) using series theory
If we define $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n \ x^{2n+1}}{(2n+1)!}$$
How to find the roots of $\sin(x)$, i.e.
$$\pi =4 \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$
satisfies $\sin (\pi)=0$
- 21
2
votes
3
answers
147
views
Infinite Series : $1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\ldots$
I need Help to evaluate infinite series : $$S=1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\cdot\cdot\cdot$$
My try:
Let $$c_n:= \left({\frac{1-i}{2}}\right)(i)^n+\left({\frac{1+i}{2}}\right)(-i)^n$$
...
- 2,322
2
votes
3
answers
104
views
Find the domain of convergence of $\sum\limits_{n=1}^{\infty} (e - (1+\dfrac{1}{n})^n)^{2x}$
I would like to find the domain of convergence of the series $\sum\limits_{n=1}^{\infty} \left(e - \left(1+\dfrac{1}{n}\right)^n\right)^{2x}$.
In fact, I knew that $\lim \left(e - \left(1+\dfrac{1}{n}...
- 71
0
votes
1
answer
65
views
I need help to evaluate series :$\sum^{\infty}_{n=0} \frac{(n+1)^{n-1}}{n!}(xe^{-x})^n$
I need help to evaluate series :$$S(x)=\sum^{\infty}_{n=0} \frac{(n+1)^{n-1}}{n!}(xe^{-x})^n$$
My attempt was to find an integral equal to $\frac{(n+1)^{n-1}}{n!}$
But I couldn't find it
This is what ...
- 2,322
2
votes
0
answers
32
views
Definition and Use of the Schett Polynomial in the Jacobi Taylor Series
I am having a tough time understanding the definition and use of the Schett polynomial introduced in the paper here. I have two questions related to this polynomial. My first question concerns its ...
- 127
4
votes
3
answers
159
views
Where is the error in evaluating this series? $\sum_{n=1}^{\infty}\frac{1}{n(4n^2-1)^2}$
Where is the error in evaluating this series? $$\sum_{n=1}^{\infty}\frac{1}{n(4n^2-1)^2}$$
We have
$$
\frac{1}{n(4n^2-1)^2}
= \frac{1}{n} - \frac{1}{2n-1} - \frac{1}{2n+1}
+ \frac{1}{2(2n-1)^2} - \...
- 2,322
2
votes
2
answers
116
views
How can to find to infinite series ? $\frac{1}{x} -\frac{1}{1!}\frac{1}{x+1}+\frac{1}{2!}\frac{1}{x+2}-\frac{1}{3!}\frac{1}{x+3}+...+\infty$
How can I continue to find a solution to infinite series ? $\frac{1}{x} -\frac{1}{1!}\cdot\frac{1}{x+1}+\frac{1}{2!}\cdot\frac{1}{x+2}-\frac{1}{3!}\cdot\frac{1}{x+3}+\cdot\cdot\cdot+\infty$
we have
$$...
- 2,322
1
vote
1
answer
87
views
Inverse Series of $x\sin x$
For some background before I get into my question: I am a Calculus 2 student who knows only some of the bare essentials to Complex Analysis, so bear with me.
I was recently studying the transformation ...
- 33
2
votes
1
answer
110
views
Closed form for sum of series with exponents in a geometric progression
I want to find the sum of this series in terms of $x$ and $n$.
$$\sum_{r=0}^{n-1}{x^{4^r}} = x + x^4 + x^{16} + ... + x^{4^{(n-1)}}$$
I can't really think of a way to approach this. I think this could ...
- 127
0
votes
1
answer
77
views
Please help me to find the sum of an infinite series. [duplicate]
Please help me to solve this problem. I need to find the sum of an infinite series:
$$
S = 1 + 1 + \frac34 + \frac12 + \frac5{16} + \cdots
$$
I tried to imagine this series as a derivative of a ...
- 21
2
votes
2
answers
80
views
Evaluate $\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$
We want to evaluate the series: $$\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$$
My try :
We have :
$$\frac{9k-4}{3k(3k-1)(3k-2)}=\frac{1}{3k-1}+\frac{1}{3k-2}-\frac{2}{3k}$$
Therefore:
$$\sum_{k=1}^...
- 2,322
5
votes
2
answers
466
views
Writing sums as integrals
There are many proofs of the Basel problem (see this wonderful thread Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)), many of which require a step ...
- 1,662