All Questions
Tagged with summation power-series
362
questions
0
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What are the four last numbers in the series $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$?
What are the four last numbers in $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$
Hello, I have come across this question, and I have no idea how to solve it. What do you guys think?
- 1,057
-1
votes
1
answer
2k
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Finite power series [duplicate]
I'm a student and I'm looking for a solution for the following finite power series:
$$
\sum_{n=0}^m \frac{1}{n!} x^n
$$
By "solution" I meant expansion of the series and finding a closed form ...
- 113
5
votes
5
answers
12k
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Formula for $r+2r^2+3r^3+...+nr^n$ [duplicate]
Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
- 3,067
4
votes
2
answers
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The autocovariance function of ARMA(1,1)
So I am reading Brockwell and Davis introduction to Time Series analysis on page 89 where he derives the ACVF of an $ARMA(1,1)$ given by:
$X_t - \phi X_{t-1}=Z_t+\theta Z_{t-1}$ with ${Z_t}$ is $WN(0,...
- 455
3
votes
1
answer
130
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Changing order of summation including a min in the summation
Lets say I have the following expression:
$$
h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x)
$$
Now my goal is to have the $v$ ...
- 1,957
1
vote
1
answer
172
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Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)
Determine the radius of convergence of the following power series:
$\sum_{n=1}^\infty n^{n^{1/3}}z^n$
Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So $z<\frac{...
- 387
1
vote
1
answer
141
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How do I extrapolate summation notation from a given series?
I am currently working on the power series for a homework assignment. I have to find the radius of convergence for the function
$$\frac{10}{1+64x^2}$$
By setting up the
$$\frac{1}{1+64x^2}$$
part ...
- 167
0
votes
2
answers
462
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Proof for multiplication of two power series
Prove that $(\sum_{k=0}^\infty u^k)^2=\sum_{k=0}^\infty (k+1)u^k$ when |u|<1.
This is a proof I need for a larger proof I was doing. I am stuck on this, so I was not able to make any notable ...
- 211
3
votes
2
answers
394
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A "generalized" exponential power series
I'm wondering if
$$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$
what would this be
$$ \sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x) $$
for $\alpha \in (0,1)$?
...
- 427
0
votes
1
answer
45
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Formula for weighed geometric sum
I'm trying to find an easy way to derive a formula for:
$S_{n} = \frac{1}{n}\sum_{i=0}^{n}(n-i)x^{i}$
I've found a recurrence relationship of sorts:
$S_{n+1} = \frac{xnS_{n}+n+1}{n+1} = x\frac{n}{n+...
- 125
2
votes
0
answers
63
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How to manipulate this summation in the easiest way possible?
$$
D =
\sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f \...
- 1,957
1
vote
3
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276
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How to prove $\sum_{k=0}^{\infty}k^2x^{k} = \frac{x(1+x)}{(1-x)^3}\text{, }|x| < 1$? [duplicate]
How do I prove that the summation $$\sum_{k=0}^{\infty}k^2x^{k} = \dfrac{x(1+x)}{(1-x)^3}\text{, }|x| < 1\text{?}$$
- 113
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0
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Question about multiplying summations with another summation inside
I have the following:
$$
y = \sum_{n=0}^\infty [x^n \sum_{k=0}^\infty (k+1)a_{k+1} P_{n-k}] \sum_{n=0}^\infty x^n[s_n - \sum_{k=0}^n a_{k+1}(k+1)R_{n-k}]
$$
I can easily multiply
$$
\sum_{n=0}^\...
- 1,957
2
votes
3
answers
220
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Simplify the sum $ \sum_{k=1}^{\infty} (\frac{1}{2})^kk $
I need some help simplifying this sum:
$$ \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^kk $$
I have a feeling it's some basic series thing that I'm forgetting, but I need help nonetheless.
- 245
3
votes
1
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Prove the series converges uniformly at $[x_0, \infty)$
Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
- 2,646