All Questions
Tagged with summation power-series
362
questions
1
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3
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131
views
How to prove that $\sum _{n=0}^{\infty }\:\frac{(x^n)'}{(n-1)!} = e^{x}(x-1)$
I am trying to prove that $$\sum _{n=0}^{\infty }\:\frac{\left(x^n\right)'}{\left(n-1\right)!} = e^{x}(x+1)\tag 1$$
This sum is very similar to the derivative of exponential $(e^x)' = \sum _{n=0}^{\...
- 23
0
votes
1
answer
66
views
Double Summation With Linked Indexes
There is this summation: $$\sum_{i=10}^{n+10}\sum_{j=i}^{n+10} j$$
and the answer is: $$\frac{1}{3} (n+1) (n+2) (n+15)$$
My question is how do you go from the summation to only the equation with n's.
- 5
1
vote
1
answer
95
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Help with proving $ \sum_{k=0}^{\infty} c_{k} x^{k}=\frac{c_{0}+\left(c_{1}-A c_{0}\right) x}{1-A x-B x^{2}} $ when $c_{k}=A c_{k-1}+B c_{k-2}$
I need some help with this question. So far i've spent a few hours on it and got a few noteworthy connections as can be seen below. But I am not sure how to progress any further. I was wondering if ...
- 27
2
votes
1
answer
35
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Evaluate $\sum\limits_{k=1}^\infty\frac{k^n}{2^{k+1}}$ for any integer $n>1$ [closed]
I know this sum is an integer. But I'm interested whether there is a closed form to this sum.
- 306
0
votes
2
answers
43
views
Is there a way to express any given natural number N as $N = \sum_{i=1}^ka_i^{p_i}$
Where every $a_i$ is the minimal and $p_i$ is maximal and $k$ (the number of terms) is the minimal, where $a_i$, $p_i$, $k$ are all natural numbers.
Examples:
$10000 = 10^4$
$164 = 10^2 + 2^6$
...
1
vote
0
answers
56
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If the Infinite sum of a series is known, what is the sum of element wise product with another series?
Suppose we know the summation of some series $G(n)$ such that, $$\sum_{n=1}^{\infty}G(n)=S_1.$$
Now assume another summation $S_2$ is expressed as,
$$S_2=\sum_{n=1}^\infty G(n) e^{i\frac{2\pi}{m}n}; \...
2
votes
3
answers
45
views
Writing a sum in terms of an appropriate function
I have a solution that is expressed as a series:
$$
\sum_{k=0}^{\infty}\left[\frac{(-1)^k t^{2k+1}}{(2k+1)!}\right]\left[4^k\right]
$$
.. and would like to show it in terms of an appropriate function, ...
- 568
2
votes
1
answer
78
views
Associativity of convolution for formal power series over a ring
Let $A\ne\{0\}$ be a ring with unity and $p,q,r\in A^{\mathbf{N}}$. I want to show that
$$\sum_{j=0}^n\sum_{k=0}^jp_kq_{j-k}r_{n-j}=\sum_{k=0}^n\sum_{j=k}^np_kq_{j-k}r_{n-j}$$
for all $n\in\mathbf{N}$....
- 733
1
vote
1
answer
936
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sum of this series: $\sum_{n=1}^{\infty}(-1)^{n-1}(\frac{1}{4n-3}+\frac{1}{4n-1})$
$$\sum_{n=1}^{\infty} (-1)^{n-1} \left(\frac{1}{4n-3}+\frac{1}{4n-1}\right)$$
What I did
$$\sum_{n=1}^{\infty} (-1)^{n-1} \left(\frac{1}{4n-3}+\frac{1}{4n-1}\right)=\sum_{n=1}^{\infty}(-1)^{n-1}\...
- 91
0
votes
3
answers
41
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Show $ f(x)=\frac{x^2}{(1-x^2)^2}-\frac{3x^2}{(3-x^2)^2}, |x|<1 $ is equal to $ f(x)=\sum_{n=0}^\infty{n(1-3^{-n})x^{2n}}, |x|<R $
Let the function $f$ be a sum-function on the interval $]-1,1[$, where $R$ is the radius of convergence.
$$
f(x)=\sum_{n=0}^\infty{n(1-3^{-n})x^{2n}}, |x|<1
$$
I find that the radius of convergence ...
- 187
2
votes
1
answer
114
views
Summation involving Gamma function
How do I prove the following?-
$\sqrt{2}\sum_{n=0}^{\infty} \frac{\Gamma(2n+1/2)(-at)^n}{n!\Gamma(n+1/2)}$=$\frac{\sqrt{1+\sqrt{1+4at}}}{\sqrt{1+4at}}$.
I think the way to obtain the right-hand side ...
- 195
2
votes
1
answer
116
views
Closed formula for the sum $a^1+a^4+a^9...$
I'm wondering if there is a closed formula for the sum $a^1+a^4+a^9...$ and more generally $a^{1^n}+a^{2^n}+a^{3^n}...$ for real $a$ and $n$ such that $|a|<1$ and $n>1$.
- 73
2
votes
1
answer
67
views
Convergence of the series $\sum_{n=1}^k\frac{k(n+1)}{k-n+1}\theta^{n}$
I am trying to show that this inequality holds
$$
\sum_{n=1}^k\frac{k(n+1)}{k-n+1}\theta^{n}<\frac{c}{(1-\theta)^2},\forall k>0,\theta\in(0,1)
$$
where $c$ is some constant. I have done some ...
- 111
1
vote
0
answers
81
views
Prove that $\sum_n 2^nx^n=\frac{1}{1-2x}$
My text book states that:
$$\sum_n 2^nx^n=\frac{1}{1-2x}$$
However this doesn't look obvious to me and I would like to prove it, but I don't know how. Could someone help me?
Background: This "...
- 2,067
3
votes
2
answers
60
views
Closed form expression for sequence of values created by differently signed series
Consider a sequence of terms of powers of $m\in\mathbb{R}$ as
$$
M_n = m^0, m^1, m^2, m^3, \ldots, m^n
$$
and create a set that contains all the values of the various signed combinations of these ...
- 3,105