All Questions
19
questions
3
votes
2
answers
117
views
Why does this proof work: $\sum\limits_{n=1}^ \infty \left(\frac{1}{4n-1} - \frac{1}{4n}\right)= \frac{\ln(64)- \pi}{8}$?
$$f(x):= \sum_{n=1}^ \infty \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$
$$f'(x) = \sum_{n=1}^ \infty ( x^{4n-2}- x^{4n-1})= \frac{x^2}{(1+x)(1+x^2)}$$
$$\int_0 ^1 \frac{x^2}{(1+x)(1+x^2)}=...
- 6,620
5
votes
1
answer
183
views
How to rigorously prove that $\sum\limits_{n=1}^ \infty( \frac{1}{4n-1} - \frac{1}{4n} )=\frac{\ln(64)- \pi}{8}$?
How to rigorously prove that $\sum\limits_{n=1}^ \infty\left( \frac{1}{4n-1} - \frac{1}{4n}\right) =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \left(\frac{x^{4n-1}}{4n-1} - \frac{x^...
- 6,620
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
1
vote
0
answers
50
views
$\dfrac{\mathrm{d}^n}{\mathrm{d}x^n}\dfrac{e^{ax}}{\ln(cx)}$ and summation with Stirling number of the first kind
I would like to calculate the $n$-th derivative of $\dfrac{e^{ax}}{\ln(cx)}$
I tried to calculate it in this way:
$$(fg)^{(n)}(x)=\sum_{k=0}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$$
$$\frac{\mathrm{d}^...
1
vote
0
answers
117
views
New asymptotic expression of $\zeta^{(m)}(1-s)$ involving generalized Stirling numbers of the $1$st kind
Introduction
A few years ago I calculated this:
For $x\to\infty$
$$\sum_{n=1}^{x}n^{s-1}\ln(n)^m\propto (-1)^m\left[\color{blue}{\zeta^{(m)}(1-s)}+\frac{1}{s}\sum_{j=0}^{s}\binom{s}{j}\ B_{s-j}^{+}x^j ...
3
votes
0
answers
107
views
Series representation of $n$th derivative of $x^n/(1+x^2)$
Find the nth derivative of $\frac{x^n}{1+x^2}$.
Please I need help in this. They are further asking to show that when $x=\cot y$ the nth derivative can be expressed as
$$n!\sin y\sum_{r=0}^{n}(-1)^r {...
0
votes
1
answer
82
views
Showing that $\sum_{i=1}^{\infty}2^{-k}\cos(kx)$ is continuous with a continuous derivative
Here is a Real Analysis problem I have managed to develop a partial answer to and would be interested in the continuation:
Question
Consider the series $g(x)=\sum_{i=1}^{\infty}$$2^{-k}\cos(kx)$ and ...
- 4,331
0
votes
2
answers
106
views
Interchange the order of unordered summation and differentiation
Let $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ and $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of a $\mathbb R$-Hilbert space $H$. Fix $x\in H$. I would like to show that $$[0,\infty)\to H\;,...
- 13.9k
5
votes
4
answers
367
views
Show that $C\frac{dC}{dr}\ + S\frac{dS}{dr}\ = (C^2 + S^2)\cos{\theta}$
Given $$C=1+r\cos{\theta}\ +\frac{r^2\cos{2\theta}}{2!}\ + \frac{r^3\cos{3\theta}}{3!}\ + \dotsb$$ and $$S = r\sin{\theta}\ + \frac{r^2\sin{2\theta}}{2!}\ + \frac{r^3\sin{3\theta}}{3!}\ + \dotsb$$
...
- 173
5
votes
1
answer
136
views
If $|f'(c)|<M$, prove $|\int_{0}^{1}f(x)dx-1/n \sum_{k=0}^{n-1}f(x/n)|<M/n$ [duplicate]
We have a derivative function $f$ with for every $c$ element of $\mathbb{R}: |f'(c)|<M$. I tried to prove that
prove that $\displaystyle \left|\int_{0}^{1}f(x)\mathrm{d}x-\frac{1}n \sum_{k=0}^{n-1}...
- 875
1
vote
1
answer
95
views
Why swapping between the derivative operator and this infinite sum leads to different results?
While working on a mathematical physical problem, i came across seemingly contradictory results.
Notations
Let's consider $\mathbf{x}_1$ to be the origin of a spherical coordinate system and $\...
0
votes
0
answers
35
views
Differentiability and Taylor series of $\sum_{k=0}^{\infty} \frac{1}{k ! \cdot\left(1+4^{k} x^{2}\right)}$.
Show, that $$g(x):=\sum_{k=0}^{\infty} \frac{1}{k ! \cdot\left(1+4^{k} x^{2}\right)}$$ is infinitely often differentiable on $\mathbb{R}$. Furthermore prove that the Taylor Series of $g$ evaluated at ...
- 73
2
votes
0
answers
272
views
Show that $(fg)^{(n)}(x)=\displaystyle\sum_{k=0}^n {n\choose k} f^{(n-k)}(x)g^{k}(x)$ [duplicate]
Let $n\geq 1,$ and let $f$ and $g$ have $n$th derivatives on $(a,b).$ Show that $(fg)^{(n)}(x)=\displaystyle\sum_{k=0}^n {n\choose k} f^{(n-k)}(x)g^{k}(x).$
My work. I think I can use induction to ...
user718615
0
votes
1
answer
579
views
Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties
For function defined as
$$
f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}
$$
check if $f$ is continuous and differentiable function.
My approach:
I would like to use the connection between this sum and ...
- 11
2
votes
1
answer
77
views
Proof that $f$ is differentiable
Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$
Proof that $f$ is differentiable on $(-2,2)$
my approach
let $ m := \frac{x}{2} $ so $m<1$
$$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
- 835