All Questions
Tagged with summation power-series
362
questions
0
votes
1
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How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $ in terms of an integral?
I have this sum
$$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $$
where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
- 267k
5
votes
1
answer
114
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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.
$f(1,n)=n$
$f'(1,n)=H_n$
$\int_0^1 \frac{f(x,...
- 267k
2
votes
1
answer
68
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I need Help to evaluate :$\sum_{n=0}^{\infty} \left({\frac{(2n+1)!!}{(2n+2)!!}}\right)^2\frac{1}{(2n+4)^2}$
I need Help to evaluate :$$S=\sum_{n=0}^{\infty} \left({\frac{(2n+1)!!}{(2n+2)!!}}\right)^2\frac{1}{(2n+4)^2}$$
we have :
$$\int^{\frac{\pi}{2}}_0\cos^{2n+2}(x)dx=\int^{\frac{\pi}{2}}_0\sin^{2n+2}(x)...
- 267k
2
votes
1
answer
70
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I need Help to evaluate series :$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$
I need Help to evaluate series :$$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$$
Let :$u_n=\frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$
We have
$$\lim_{n\to\infty} n\left({\frac{u_n}{u_{n+1}}-...
- 2,968
28
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6
answers
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Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$
I am wondering if there exists any formula for the following power series :
$$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$
Is there any way to calculate the sum of above series (if ...
- 17.4k
0
votes
1
answer
77
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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 ...
- 1,381
2
votes
2
answers
80
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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:
$$\sum_{k=1}^...
- 267k
1
vote
2
answers
100
views
Find sum of power series
The problem is to find the sum of the power series:
$$\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)(n+1)}$$
My solution:
First to find where the sum exists (for which x):
Using D'Alembert's criterion for ...
- 2,288
0
votes
0
answers
62
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Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]
I am looking for ways to simplify the sum
$$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$
The first thought I had approaching this was to use Hurwitz and/or Zeta ...
- 45
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votes
0
answers
34
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Variants of geometric sum formula
I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$
and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$.
(See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From Sum of ...
60
votes
11
answers
127k
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The idea behind the sum of powers of 2
I know that the sum of powers of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.
For example, sum of n numbers is $\...
- 61.4k
1
vote
1
answer
129
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Borel Regularization of $\sum_{n=1}^\infty \ln(n)$ [closed]
I'm trying to solve the following taylor series
$$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$
so I can regularize the following sum
$$\sum_{n=1}^\infty \ln(n)$$
Using Borel Regularizaiton I can use ...
- 16.1k
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
- 51
1
vote
1
answer
60
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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 $8.1$
- 1,961
5
votes
0
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101
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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 ...
