All Questions
Tagged with summation power-series
362
questions
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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 ...
- 834
3
votes
2
answers
143
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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 ...
- 2,369
4
votes
1
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163
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Expansion of Confluent Hypergeometric Function in terms of $\operatorname{erfi}(x)$
I have the following confluent hypergeometric function: $_1F_1\left(2(m+1),\frac{1}{2},-x^2\right)$.
By using Mathematica, I know that for values of $m=0,1,2,...$ this function expands into a power ...
- 47
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64
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Evaluation of sum of power series $\sum \frac{n}{n^2-1}x^n$
Evaluation of sum of power series $\sum \frac{n}{n^2-1}x^n$ ?
I know $\sum_{n\geq0} nx^n = x/(1-x)^2$, how do I Include the $\frac{1}{n^2-1}$ ?
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1
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3
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64
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Convergence of $\sum_{k=1}^{\infty} \sum_{u=0}^{k-1}\frac{r^{u}}{(k-u)^2}$
Assume $r \in (0,1)$, I'm looking at the following sum, conjecture that it converges.
$$ \sum_{k=1}^{\infty} \sum_{u=0}^{k-1}\frac{r^{u}}{(k-u)^2} $$
This is very similar to the result of one earlier ...
- 740
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Find the sum: $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ [duplicate]
Find the sum:
$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$$
My try:
I played a bit with the coefficient to make it look easier/familiar:
First attempt:
$$\begin{align}
\sum_{n=0}^\infty \frac{(n!)^2}{...
- 877
4
votes
6
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955
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Prove this formula $\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$
I am trying to use prove, by just simple algebraic manipulation, to prove the equality of this formula.
$$\dfrac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$$
I have ...
- 3,187
2
votes
1
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Infinite series simplification
Is there a clean way to simplify the following series:
$$
0.5^2(1-2*0.5^2)+0.5^3(1-2*0.5^3)+\cdots +0.5^k(1-2*0.5^k)
$$
where k = 1, 2, ... ∞
Using R led to the convergence point = 1/3
...
user14688199
1
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1
answer
104
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The Radius of Convergence of $\sum_{n=1}^{\infty} x^n\left( \frac{n}{2n+1} \right)^{2n-1}$
The series is in the form of power series, here $x_{0}=0$. We can apply Root Test(We've concluded to use the Root Test in my previous question).
$\displaystyle\sum_{n=1}^{\infty} x^n\left( \dfrac{n}{...
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45
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Intuition Behind Step In Power Series Solution
When solving an ODE by the power series method, one ends up with an expression multiplied by $x^n$ inside a summation equaling $0$, and the next step is to set that expression equal to $0$. For ...
- 1,912
6
votes
2
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263
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Hard power series problem
Consider the differential equation $$(1+t)y''+2y=0$$
with the variabel coefficient $(1+t)$, with $t\in \mathbb{R}$.
Set $y(t)=\sum_{n=0}^{\infty}a_nt^n$. What are the first 4 terms in the associated ...
- 539
0
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2
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93
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Manipulation of summation index - power series
Problem
Consider the differential equation: $y''+ty'+2y=t^2e^t $
By setting $y(t)=\sum_{n=0}^{\infty}c_nt^n$ show that the differential equation can be rewritten as
$$\sum_{n=0}^{\infty} \big((n+2)(n+...
- 539
0
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0
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174
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find the sum of infinite series $\sum_{n=0}^{\infty}nx^n$ [duplicate]
How to find the sum of infinite series $\sum_{n=0}^{\infty}nx^n$? I have no clue where to begin with.
- 560
1
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1
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171
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Convert to a compact form
I'm trying to help someone regarding Maclaurin Series. It's been a few years since i've done formal maths, and (in fact i used python to compute the derivatives) the series is
$$ 2x-\frac{8}{3}x^3+\...
1
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3
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it's possible to invert summation/ series limits?
If the summation just sum every term i was thinking that for instance 1+2+3+4 = 4+3+2+1
so why this $$\sum\limits_{i=1}^{n} (3i)\ = \sum\limits_{i=n}^{1} (3i) $$ is not true ?
And how i can ...
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