All Questions
11
questions
6
votes
0
answers
104
views
Please help me identify any errors in my solution to the following DE: $xf(x)-f'(x)=0$, $f(0)=1$
Context/background:
I am self-studying series, first in the context of generating functions and now in the context of functional/differential equations. As such, I like to set myself practise problems,...
- 1,632
6
votes
1
answer
275
views
Closed form for $\int \sqrt[n]{\tan x}\ dx$
I was solving $\displaystyle\int\sqrt[n]{\tan x}\ dx$.
Here's my method:
$$\begin{align}\int\sqrt[n]{\tan x}\ dx &= \int\frac{n \cdot t^n}{1 + (t)^{2n}}\tag{1}\ dt\\& = n \int\sum_{k=0}^\infty ...
- 1,614
2
votes
1
answer
659
views
What function does $\sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)n!^24^n}\frac{x^n}{y^{2n}}$ correspond to?
I have the following Taylor series, and although it looks familiar, I cannot figure out which function it corresponds to! Does anyone recognize this Taylor series?
$$1+\frac{x}{2y^2} - \frac{x^2}{8y^...
- 149
13
votes
1
answer
549
views
How to prove this series about Fibonacci number: $\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$? [duplicate]
How to prove this series result:
$$\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$$
where $F_{1}=1,~F_{2}=1,~F_n=F_{n-1}+F_{n-2},~~n\geq 3$.
I have no idea where to start.
- 5,281
9
votes
1
answer
416
views
Closed Form for $~\lim\limits_{n\to\infty}~\sqrt n\cdot(-e)^{-n}\cdot\sum\limits_{k=0}^n\frac{(-n)^k}{k!}$
$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ?
Inspired by this frequently asked question, I ...
- 48.5k
1
vote
2
answers
70
views
Closed form of a power series
Find the function that represents the following sum: $\sum\limits_{k=0} ^\infty \frac{(n^2)}{n!} x^n$. Can't find the function that represents this.
- 41
0
votes
0
answers
878
views
Series identity for cotangent
How to prove that $x \cot(x) = 1 - 2 \sum_{n=0}^{\infty}{\frac{x^{2}}{(n \pi)^{2}-x^{2}}}$?
First, it does not seem to be solvable, using considerations regarding Taylor series. The Fourier approach ...
- 2,951
6
votes
3
answers
131
views
what's the summation of this finite sequence?
$a$ and $b$ are positive integers.
The summation is
$$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}}
{a + b - x}\\
b
\end{array}} \right)} .$$
Any closed-form expression?
I thought it ...
- 1,527
2
votes
1
answer
178
views
summation of a finite sequence?
What is the summation of the finite sequence:
$$\sum\limits_{i = 1}^n {\frac{1}{i}\left( {\begin{array}{*{20}{c}}
{2i - 2}\\
{i - 1}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{2n + 2 - 2i}\\
...
- 1,527
5
votes
1
answer
174
views
what is the summation of such a finite sequence?
The summation is: $$\sum_{i=0}^n \binom{2i}i \binom{2n-2i}{n-i}$$
The answer is $4^n$.
How to prove it, and how to think out it?
- 1,527
2
votes
3
answers
264
views
Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $
I was wondering if there is an explicit formulation for the series
$$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$
It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?
- 2,433