All Questions
Tagged with summation power-series
362
questions
0
votes
1
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221
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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
2
votes
0
answers
103
views
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
answer
129
views
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
views
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
answers
441
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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
vote
0
answers
46
views
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
vote
0
answers
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: $$ \sum_{n=0}^{\infty} a_{n+2}(n+1)(n+2)x^{n} - 2\sum_{n=0}^{\infty} a_{n}nx^{n} + \...
- 135
1
vote
1
answer
69
views
Determine if the series representation is true or not
A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in ...
- 79
0
votes
2
answers
52
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What is the Radius of Convergene of $f(z) = \sum_{n=0}^\infty \frac1{4^n}z^{2n+1}$
Here's what I have:
$f(z) = z + \frac14z^3+\frac1{4^2}z^5+...$
So, my coefficients are either $0$ or $\frac1{4^n}$ with $\frac1{4^n}$ being the supremum.
So, $\limsup \limits_{n \to \infty} |c_n|^{\...
- 70
0
votes
1
answer
57
views
Find the radius of convergence of $\sum_{i=0}^\infty a_n$ where $\sum_{i=0}^\infty 2^n a_n$ converges, but $\sum_{i=0}^\infty (-1)^n2_na_n$ diverges [closed]
From this example: $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty|a_n|$ diverges. Then Radius of convergence?
I believe I'm supposed to leverage these two statements to show that $R \leq |z|$...
- 70
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votes
1
answer
45
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Show $\liminf_n |\sum^n a_kb_{n-k}|^{-\frac{1}{n}} = \liminf_n |a_n|^{-\frac{1}{n}}$?
Suppose for some complex $a_n$ and $b_n$ such that $\liminf_n |a_n|^{-\frac{1}{n}} = \liminf_n |b_n|^{-\frac{1}{n}}= R$. Show that $\liminf_n |\sum^n a_kb_{n-k}|^{-\frac{1}{n}} = R$?
I'm somehow ...
- 834
3
votes
2
answers
143
views
Evaluate infinite s, series, similar to $\cos(z)$
Evaluate the sum
$$\sum_{k=0}^\infty\frac{(-1)^kz^{ak}}{\Gamma(1+ak)}$$
where $a\in \mathbb{R}_{>0}$ and $z\in\mathbb{C}$.
I know if $a=2$ then this is the series expansion for $\cos(z)$. But for ...
- 2,365
4
votes
1
answer
164
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Expansion of Confluent Hypergeometric Function in terms of $\operatorname{erfi}(x)$
I have the following confluent hypergeometric function: $_1F_1\left(2(m+1),\frac{1}{2},-x^2\right)$.
By using Mathematica, I know that for values of $m=0,1,2,...$ this function expands into a power ...
- 47
0
votes
1
answer
64
views
Evaluation of sum of power series $\sum \frac{n}{n^2-1}x^n$
Evaluation of sum of power series $\sum \frac{n}{n^2-1}x^n$ ?
I know $\sum_{n\geq0} nx^n = x/(1-x)^2$, how do I Include the $\frac{1}{n^2-1}$ ?
- 1
1
vote
3
answers
64
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Convergence of $\sum_{k=1}^{\infty} \sum_{u=0}^{k-1}\frac{r^{u}}{(k-u)^2}$
Assume $r \in (0,1)$, I'm looking at the following sum, conjecture that it converges.
$$ \sum_{k=1}^{\infty} \sum_{u=0}^{k-1}\frac{r^{u}}{(k-u)^2} $$
This is very similar to the result of one earlier ...
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10
votes
2
answers
2k
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Find the sum: $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ [duplicate]
Find the sum:
$$\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
answers
962
views
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
answer
97
views
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
vote
1
answer
105
views
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}{...
0
votes
0
answers
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 ...
- 1,922
6
votes
2
answers
264
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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
votes
2
answers
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
$$\sum_{n=0}^{\infty} \big((n+2)(n+...
- 539
0
votes
0
answers
174
views
find the sum of infinite series $\sum_{n=0}^{\infty}nx^n$ [duplicate]
How to find the sum of infinite series $\sum_{n=0}^{\infty}nx^n$? I have no clue where to begin with.
- 560
1
vote
1
answer
171
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Convert to a compact form
I'm trying to help someone regarding Maclaurin Series. It's been a few years since i've done formal maths, and (in fact i used python to compute the derivatives) the series is
$$ 2x-\frac{8}{3}x^3+\...
1
vote
3
answers
1k
views
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 ...
- 241
1
vote
3
answers
131
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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}^{\...
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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
views
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
views
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}$$
for all $n\in\mathbf{N}$....
- 733
1
vote
1
answer
936
views
sum of this series: $\sum_{n=1}^{\infty}(-1)^{n-1}(\frac{1}{4n-3}+\frac{1}{4n-1})$
$$\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
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votes
3
answers
41
views
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
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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
1
vote
1
answer
49
views
Show that $F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$
I've been working on a recent exercise question where I was asked to show that:
$$F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$$
Now I cansee that the infinite sum is ...
- 115
1
vote
0
answers
36
views
Field of convergence, prove it's open?
For the following power series $$ \sum_{n=1}^{ \infty} \bigg( 1 + \frac{(-1)^n}{n} \bigg)^{ n^2} \frac{(2x+1)^n}{n} $$ I proved that the radius of convergence is $r= \frac{1-e}{2 \ e} $.
How may I ...
- 421
0
votes
3
answers
31
views
an infinite series that i couldn't figured out how to sum up
I have the series which is $$\sum_{n=0}^{\infty} 2^{-n(x-1)}$$ and from the ratio test it converges for all $x\geq 2$ but how can i find the general sum of the series wrt $x$
- 114
1
vote
1
answer
540
views
Calculating sum of series using derivative of a function
We're given the following problem:
"We know that $\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k $ for $ -1 < x < 1 $. Using the derivative with respect to $x$, calculate the sum of the following ...
- 181
9
votes
1
answer
1k
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Proving $\pi=(27S-36)/(8\sqrt{3})$, where $S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$ [closed]
I have to prove that:
$$\pi=\frac{27S-36}{8\sqrt{3}}$$
where I know that $$S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$$
Where do I get started?
- 125
0
votes
1
answer
65
views
Computing the sum of an alternating power series
I'm asked to find the sum of a power series
for all x in the interior of the domain of convergence, which I found to be [-1,1]. The question also gives the hint to take the second derivative of the ...
- 41
0
votes
2
answers
62
views
Show that a the sum function of a power series is differentiable twice and that $f''(x) = \frac{1}{49-7x}$
I am studying for my analysis exam and are to consider the power series
$$
\sum_{n=2}^\infty \frac{1}{n(n-1)7^n}z^n
$$
with the sum function for $x \in ]-7,7[$ given by
$$
f(x) = \sum_{n=2}^\infty \...
- 917
0
votes
2
answers
69
views
Infinite sum power series
I would like to show
$$
\sum_{r=0}^{\infty}\frac{1}{6^r} \binom{2r}{r}= \sqrt{3}
$$
I have tried proving this using telescoping sum, limit of a sum, and some combinatorial properties but I couldn't ...
- 41
1
vote
2
answers
461
views
Sum of finite series using partial fraction
I'm quite stuck with the following problem. I have seen on this forum that there is already an answer for the infinite sum to the problem but I can't seem to find how to find the sum for a finite ...
- 181
0
votes
1
answer
49
views
Theoretical Procedure for Power Series Equation:
If I have the following equation: \begin{equation}2c_0(x-1)+\sum_{k=2}^\infty[(c_{k-2}+2c_{k-1})(x-1)^k]+\sum_{k=0}^\infty[(c_{k+2}(k+2)(k+1)+kc_k+(k+1)c_{k+1}+c_k)(x-1)^k]=0 \end{equation}
I was ...
- 2,653