All Questions
Tagged with summation power-series
362
questions
5
votes
2
answers
498
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\...
- 2,756
5
votes
1
answer
104
views
Determine whether or not $\exp\left(\sum_{n=1}^{\infty}\frac{B(n)}{n(n+1)}\right)$ is a rational number
Let $B(n)$ be the number of ones in the base 2 expression for the positive integer n.
Determine whether or not $$\exp\left(\sum_{n=1}^{\infty}\frac{B(n)}{n(n+1)}\right)$$ is a rational number.
...
- 177
5
votes
2
answers
598
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Generalization of Faulhaber's formula
Is there a way to calculate a sum of non-integer positive powers of integers?
$\sum_{k=1}^nk^p: n \in \mathbb{N}, p \in \mathbb{R^+}$
There's a Faulhaber's formula, but as far as I can see, it is ...
- 230
5
votes
1
answer
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Studies about $\sum_{k=1}^{n} x^{\frac 1k}$
Are there any studies about this function?
$$f(x,n)=\sum_{k=1}^{n} x^{1/k}=x+x^{1/2}+x^{1/3}+x^{1/4}+\cdots +x^{1/n}$$
EDIT:
My first notes about it.
$f(1,n)=n$
$f'(1,n)=H_n$
$\int_0^1 \frac{f(x,...
- 611
5
votes
1
answer
293
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Solving 2nd order ODE with Frobenius method - problems with summation symbol
I'm trying to solve the ODE:
$$ y''(x) + \frac{2x}{(x-1)(2x-1)} y'(x) - \frac{2}{(x-1)(2x-1)} y(x) = 0 $$
I'm trying to find a solution by the Frobenius method, expanding a power series of the ...
- 81
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0
answers
102
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Summing a nonstandard sequence, closed form of $S_n(x) = \sum_{i=1}^n x^{c^{i-1}}$
Arithmetic sequences have a common difference, where you add a constant to each term to get the next. Geometric sequences have a common ratio, where you multiply a constant to each term to get the ...
5
votes
0
answers
113
views
On the sum of Double Power Series $\sum\limits_{-\infty}^{+\infty}\space\frac{2^n}{{x}^{2^n}+1}$ [duplicate]
Let $\,{x\in\mathbb{C}}\,$, show that:
$$ S_{\small-}=\sum_{n=1}^{\infty}\space\frac{2^{-n}}{{x}^{2^{-n}}+1}\,=\frac{1}{\log{x}}-\frac{1}{x-1}\qquad\qquad\qquad\tag{1} $$
$$ S_{\small+}=\sum_{n=...
- 3,690
4
votes
4
answers
2k
views
Index change of variable with product
This is probably a dumb question, but oh well here it is:
One day a student came to me and asked: Why isn't
$$
\sum_{k=0}^{\infty} \frac{1}{2^{2k}}=\sum_{n=0}^{\infty} \frac{1}{2^{n}}$$
using a ...
- 4,623
4
votes
6
answers
963
views
Prove this formula $\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$
I am trying to use prove, by just simple algebraic manipulation, to prove the equality of this formula.
$$\dfrac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$$
I have ...
- 3,187
4
votes
5
answers
338
views
Power series summation [closed]
Trying to find the sum of the following infinite series:
$$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$
Any ideas on how to find this sum?
- 277
4
votes
3
answers
159
views
Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$
Let $0<p<1$,Find the sum
$$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
- 93.6k
4
votes
2
answers
223
views
Methods for finding and guessing closed forms of infinite series
I want to prove $\displaystyle\sum_{k \ge 0} \Big(\frac{1}{3k+1} - \frac{1}{3k+2}\Big) = \frac{\pi}{\sqrt{27}}$
The reason for this question is I was doing the integral $\displaystyle\int_0^{\infty} \...
- 1,863
4
votes
2
answers
299
views
Transfer between integrals and infinite sums
So I was watching a video on YouTube about how $$\sum_{i=1}^\infty \frac{\chi(i)}{i} = \frac{\pi}{4}$$ (note that $\chi(i) = 0$ for even numbers $i$, $1$ for $\text{mod}(i, 4) = 1$, and $-1$ for $\...
- 2,402
4
votes
5
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161
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Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$
Find the sum of $$1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$$
a) $\dfrac{\pi}8(\sqrt2-1)$
b) $\dfrac{\pi}4(\sqrt2-1)$
c) $\dfrac{\pi}8(\sqrt2+1)$
d) $\dfrac{\pi}4(\sqrt2+1)$
...
- 45
4
votes
4
answers
187
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What is the sum of $\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$?
What is the sum of the following expression:
$$\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$$
I know it is convergent but I cannot evaluate its sum.
- 24.5k