All Questions
Tagged with summation power-series
362
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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.
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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 ...
- 178k
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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?
- 268k
6
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2
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Hard power series problem
Consider the differential equation $$(1+t)y''+2y=0$$
with the variabel coefficient $(1+t)$, with $t\in \mathbb{R}$.
Set $y(t)=\sum_{n=0}^{\infty}a_nt^n$. What are the first 4 terms in the associated ...
- 5,691
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3
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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.
- 7,792
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3
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Computing:$\sum_{n=0}^\infty\frac{3^n}{n!(n+3)}$
I'm learning the subject Power Series and I can't figure out how to find the sum of the series $$\sum_{n=0}^\infty\frac{3^n}{n!(n+3)}$$.
I know that the power series $\sum_{n=0}^\infty\frac{3^n}{n!(n+...
- 3,348
2
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2
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442
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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 ...
- 109k
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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 ...
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1
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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: $$ \sum_{n=0}^{\infty} a_{n+2}(n+1)(n+2)x^{n} - 2\sum_{n=0}^{\infty} a_{n}nx^{n} + \...
- 135
1
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1
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Determine if the series representation is true or not
A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in ...
- 40.1k
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2
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What is the Radius of Convergene of $f(z) = \sum_{n=0}^\infty \frac1{4^n}z^{2n+1}$
Here's what I have:
$f(z) = z + \frac14z^3+\frac1{4^2}z^5+...$
So, my coefficients are either $0$ or $\frac1{4^n}$ with $\frac1{4^n}$ being the supremum.
So, $\limsup \limits_{n \to \infty} |c_n|^{\...
- 40.8k
3
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2
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Evaluate infinite s, series, similar to $\cos(z)$
Evaluate the sum
$$\sum_{k=0}^\infty\frac{(-1)^kz^{ak}}{\Gamma(1+ak)}$$
where $a\in \mathbb{R}_{>0}$ and $z\in\mathbb{C}$.
I know if $a=2$ then this is the series expansion for $\cos(z)$. But for ...
- 43.5k
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1
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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 $R \leq |z|$...
- 77.5k
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Show $\liminf_n |\sum^n a_kb_{n-k}|^{-\frac{1}{n}} = \liminf_n |a_n|^{-\frac{1}{n}}$?
Suppose for some complex $a_n$ and $b_n$ such that $\liminf_n |a_n|^{-\frac{1}{n}} = \liminf_n |b_n|^{-\frac{1}{n}}= R$. Show that $\liminf_n |\sum^n a_kb_{n-k}|^{-\frac{1}{n}} = R$?
I'm somehow ...
- 40.8k
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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,...
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