All Questions
Tagged with summation power-series
362
questions
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58
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Can we reduce the 3-nested summation into 2-nest summation?
My problem started with the first case that I have :
$$
I_{2}=a^{k_1} (a+b)^{k_2}
$$
Where $k_1,k_2$ are real positive integers. Using the series expansion :
$$
(a+b)^n=\sum_{i_1=0}^{n} \binom{n}{i_1} ...
- 45
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0
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65
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How to find the sum of a power series without knowing the actual power series
How do I find the sum of this series?
$$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} $$
The approach I wanted to use is to find a power series that can become this number series for a certain ...
- 11
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votes
1
answer
71
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Evaluate $\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$
Evaluate $$\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$$ using the fact that
$${\frac{n}{n^2-1}} = {\frac{1}{2(n-1)}} + {\frac{1}{2(n+1)}}$$
So far I have proven that the Radius of Convergence is 1 and ...
- 63
1
vote
2
answers
121
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Upper bound for infinite sum
Let $x \in (0,1)$, I need to find an upper bound (as good as possible) for the series
$$\sum_{n=1}^{\infty}x^n n^k,$$
where $k$ is a natural number larger or equal than $2$, i.e., $k=2,3,4,\dots$.
My ...
- 35.9k
2
votes
1
answer
148
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How to expand the product of Laguerre polynomials into a sum of series？
In the course of my research, I needed a formula and found it, but I can not understand the derivation process of the formula. How to extract the $t^n$ and get the $\theta(m-p)$ in the last step? Can ...
- 145
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votes
1
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217
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Closed form of sum of n^n series? [closed]
Hi readers, may i know how this formula is derived? I thought of using https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula which is Euler-Maclaurian to find the formula but its not.
- 51
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103
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What is an approximate closed form for sum of $n^n$ series?
I read about this https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula which is Euler-Maclaurain and I did my calculation to find the constants. However, the answer differ greatly from ...
- 51
-1
votes
1
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127
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Sum of 1.5-powers of natural numbers using Euler-Maclaurin Formula?
Hi, I have read this Sum of 1.5-powers of natural numbers and answers posted by Mark Viola. However, I am not too sure on how he got to this.
Can anyone show me as I am quite new to series?
- 51
1
vote
3
answers
171
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Find explicit formula for summation for p>0
Find explicit formula for summation for any $p>0$:
$$\sum_{k=1}^n(-1)^kk\binom{n}{k}p^k.$$
Any ideas how to do that? I can't figure out how to bite it. I appreciate any help.
- 33
2
votes
2
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437
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Nested Sum of nested series
I came across this series while doing a problem today,
$$\sum_{k=0}^\infty\left(\sum_{n=0}^ka_n\right)x^k-\left(\sum_{n=0}^k x^{n}\right)a_k$$
And I wasnt able to get any further with it, but thought ...
- 21
1
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0
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46
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Is this summation equality true
A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in ...
- 79
1
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0
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101
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Frobenius Method solution and radius of convergence
I was given the equation $$y''-2xy'+\mu y = 0 $$ where $\mu$ is a parameter $\geq{0}$.
I got the relation of sums: $$ \sum_{n=0}^{\infty} a_{n+2}(n+1)(n+2)x^{n} - 2\sum_{n=0}^{\infty} a_{n}nx^{n} + \...
- 135
1
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1
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69
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Determine if the series representation is true or not
A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in ...
- 79
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2
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52
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What is the Radius of Convergene of $f(z) = \sum_{n=0}^\infty \frac1{4^n}z^{2n+1}$
Here's what I have:
$f(z) = z + \frac14z^3+\frac1{4^2}z^5+...$
So, my coefficients are either $0$ or $\frac1{4^n}$ with $\frac1{4^n}$ being the supremum.
So, $\limsup \limits_{n \to \infty} |c_n|^{\...
- 70
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1
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57
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Find the radius of convergence of $\sum_{i=0}^\infty a_n$ where $\sum_{i=0}^\infty 2^n a_n$ converges, but $\sum_{i=0}^\infty (-1)^n2_na_n$ diverges [closed]
From this example: $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty|a_n|$ diverges. Then Radius of convergence?
I believe I'm supposed to leverage these two statements to show that $R \leq |z|$...
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