All Questions
Tagged with summation power-series
362
questions
2
votes
1
answer
68
views
I need Help to evaluate :$\sum_{n=0}^{\infty} \left({\frac{(2n+1)!!}{(2n+2)!!}}\right)^2\frac{1}{(2n+4)^2}$
I need Help to evaluate :$$S=\sum_{n=0}^{\infty} \left({\frac{(2n+1)!!}{(2n+2)!!}}\right)^2\frac{1}{(2n+4)^2}$$
we have :
$$\int^{\frac{\pi}{2}}_0\cos^{2n+2}(x)dx=\int^{\frac{\pi}{2}}_0\sin^{2n+2}(x)...
- 2,288
2
votes
1
answer
70
views
I need Help to evaluate series :$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$
I need Help to evaluate series :$$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$$
Let :$u_n=\frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$
We have
$$\lim_{n\to\infty} n\left({\frac{u_n}{u_{n+1}}-...
- 2,288
0
votes
1
answer
77
views
Please help me to find the sum of an infinite series. [duplicate]
Please help me to solve this problem. I need to find the sum of an infinite series:
$$
S = 1 + 1 + \frac34 + \frac12 + \frac5{16} + \cdots
$$
I tried to imagine this series as a derivative of a ...
- 21
2
votes
2
answers
80
views
Evaluate $\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$
We want to evaluate the series: $$\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$$
My try :
We have :
$$\frac{9k-4}{3k(3k-1)(3k-2)}=\frac{1}{3k-1}+\frac{1}{3k-2}-\frac{2}{3k}$$
Therefore:
$$\sum_{k=1}^...
- 2,288
1
vote
2
answers
100
views
Find sum of power series
The problem is to find the sum of the power series:
$$\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)(n+1)}$$
My solution:
First to find where the sum exists (for which x):
Using D'Alembert's criterion for ...
- 13
0
votes
0
answers
62
views
Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]
I am looking for ways to simplify the sum
$$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$
The first thought I had approaching this was to use Hurwitz and/or Zeta ...
- 45
0
votes
0
answers
34
views
Variants of geometric sum formula
I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$
and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$.
(See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From Sum of ...
1
vote
1
answer
129
views
Borel Regularization of $\sum_{n=1}^\infty \ln(n)$ [closed]
I'm trying to solve the following taylor series
$$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$
so I can regularize the following sum
$$\sum_{n=1}^\infty \ln(n)$$
Using Borel Regularizaiton I can use ...
0
votes
1
answer
41
views
How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $ in terms of an integral?
I have this sum
$$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $$
where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
- 51
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
- 51
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
5
votes
0
answers
101
views
Summing a nonstandard sequence, closed form of $S_n(x) = \sum_{i=1}^n x^{c^{i-1}}$
Arithmetic sequences have a common difference, where you add a constant to each term to get the next. Geometric sequences have a common ratio, where you multiply a constant to each term to get the ...
1
vote
1
answer
99
views
Calculating the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$ [duplicate]
I want to find the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$.
I have tried to turn it into a power series for a known function, with no luck. I also tried to write it as $\sum_{n=0}^{\infty} \...
- 13
1
vote
1
answer
97
views
Evaluate the sum of $1/n^6$ using Euler's method
I've just learned how Euler evaluated $1+1/2^2+1/3^2+...+1/n^2+...=\pi^2/6$ by comparing the coefficients of the series form and product form of $\sin(x)/x$.
The series form is
$$\dfrac{\sin(x)}{x}=1-\...
- 61
4
votes
1
answer
88
views
A conjecture involving series with zeta function
Recently, I tried to evaluate a limit proposed by MSE user Black Emperor. In the process of evaluating the limit, I have obtained the following equality.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 1,240
2
votes
1
answer
248
views
What are the conditions for Ramanujan's Master Theorem to hold?
Ramanujan's Master Theorem states that if
$$f(x) = \sum_{k=0}^{\infty} \frac{\phi(k)}{k!}(-x)^k$$
then $$\int_{0}^{\infty}x^{s-1}f(x)\ dx = \Gamma(s)\phi(-s).$$
But there are obviously some conditions ...
- 359
0
votes
0
answers
23
views
Summation of the following form with non-integer n
I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
- 109
2
votes
3
answers
110
views
How can I derive the first two terms of the asymptotic expansion of $f(n)=\sum_{k=1}^\infty [(-1)^k/k]\ln(n^2+k)$ at $n \to +\infty$?
I am struggling with the problem in the title of this post. I have tried many different methods, but nothing has worked so far. I only managed to derive the first term of the asymptotic expansion:
$f(...
- 135
1
vote
0
answers
44
views
Multiplication of multiple summations of complex functions
I have a series that looks like $\sum_{l,m,n}\frac{A^{l}B^{m}C^{n}}{l!m!n!}$ where $A$ is a complex function and $B$ and $C$ are real functions. The summation is finite up to some cutoff $p$. $A$, $B$,...
- 21
0
votes
0
answers
39
views
Converting a power series with recursively related coefficients into a single sigma sum expression
EDIT: Ok, silly me. There is an obvious closed form summation which somehow escaped me. Nonetheless, I would appreciate comments on deriving a characteristic polynomial from the generating function.
...
- 434
0
votes
0
answers
60
views
Rewriting a sum with a floor function as upper limit
I am having some trouble in rewriting a sum whose upper limit is given in terms of a floor function $\lfloor \cdot \rfloor$. The task is to prove that both sides of the following expression coincide:
$...
- 91
4
votes
2
answers
223
views
Methods for finding and guessing closed forms of infinite series
I want to prove $\displaystyle\sum_{k \ge 0} \Big(\frac{1}{3k+1} - \frac{1}{3k+2}\Big) = \frac{\pi}{\sqrt{27}}$
The reason for this question is I was doing the integral $\displaystyle\int_0^{\infty} \...
- 1,863
4
votes
2
answers
133
views
Interchanging summations with complicated, nested indices
I have a question regarding interchanging the order of three nested summations. My expression looks like
\begin{align}
\sum_{n=0}^\infty \sum_{k=0}^n \sum_{\nu=0}^{4n-2k}\frac{C_{nk\nu}}{k!(n-k)!}\...
- 91
0
votes
0
answers
48
views
Multiplication of a power series and a finite-order polynomial [duplicate]
I am trying to find a general expression for the coefficients of the power series that results from the multiplication of a polynomial and a power series. I have looked at this post Convolution and ...
- 317
0
votes
1
answer
145
views
Solving a sum similar to geometric series
How do I solve the sum
$$\sum_{k=1}^y \left( 1-\frac{1}{\ln x} \right)^k \hspace{0.5cm} $$
for $x>0$ and $y$ a positive integer greater than one?
Despite resembling a geometric series, it does not ...
user1089309
2
votes
1
answer
167
views
Find the summation of $\sum_{n\geq1}\frac{3^n}{n\left(\frac{1}{n}+1\right)^n}x^n$
I was trying to find what the summation of $$\sum_{n\geq1}\frac{3^n}{n\left(\frac{1}{n}+1\right)^n}x^n$$ is, but I'm kind of stuck.
I recognized the pattern at the bottom as
$$\lim_{n\to+\infty}\left(...
- 41
0
votes
1
answer
76
views
Prove the formula $1+r\cdot \cos(α)+r^{2}\cos(2α)+\cdots+r^{n}\cos(nα)=\dfrac{r^{n+2}\cos(nα)-r^{n+1}\cos[(n+1)α]-r\cosα+1}{r^{2}-2r\cdot \cos(α)+1}$
For $r,a\in\mathbb{R}:\; r^{2}-2r\cos{a}+1\neq 0$ prove the formula $$1+r\cdot \cos(a)+r^{2}\cos(2a)+\cdots+r^{n}\cos(na)=\dfrac{r^{n+2}\cos(na)-r^{n+1}\cos[(n+1)a]-r\cdot \cos(a)+1}{r^{2}-2r\cdot \...
- 21
1
vote
1
answer
93
views
Expanding denominator in a power series, mismatch of the expansion
Below is a snippet from the book Ralston:First course in numerical analysis but it seems to me that something is wrong with $(10.2-11):$ the denominator divided by $a_1\lambda_1^m$ starts with $1$ not ...
- 3,978
3
votes
1
answer
135
views
An infinite sum of products
I have to calculate this sum in closed form
$$ \sum_{n=1}^\infty \prod_{k=1}^n \frac{x^{k-1}}{1 - x^k} $$
where $x < 1$.
Numerical evaluation shows that this converges. The product can be performed ...
- 86
6
votes
2
answers
115
views
Proving that the exponential satisfies the following sum equation
I was thinking about how $(\sum_{n=0}^{\infty} \frac{1}{n!})^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
$
and was wondering if there existed any other sequences that satisfied this besides the exponential....
- 506
0
votes
1
answer
42
views
If $f(x) = \sum_{n=1}^\infty \frac{2^n}{n}x^n$ for $x \in ]-\frac{1}{2}, \frac{1}{2}[$ then what is $f'(x)$?
If $$f(x) = \sum_{n=1}^\infty \frac{2^n}{n}x^n \: \: \text{for}\: \: x \in ]-\frac{1}{2}, \frac{1}{2}[$$ then what is $f'(x)$?
Attempt
It turns out that $\rho = \frac{1}{2}$ is the radius of ...
- 539
1
vote
0
answers
56
views
How to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})\;$ as a linear function?
Suppose we have a finite set $I \subseteq \mathbb{N}$, and $\alpha_i\in [0, 1]$ are fixed numbers for all $i\in I$.
Is there a way to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})...
- 342
-1
votes
1
answer
68
views
A closed form for the sum of a series $\sum_{n=1}^{\infty}x^{n\alpha} /\Gamma{(n \alpha)}$
Let $\alpha \in (0,1)$. Is there a closed form for the sum $\sum_{n=1}^{\infty}x^{n\alpha}/\Gamma{(n \alpha)} $ ?
- 3,165
0
votes
0
answers
105
views
Manipulation with the following infinite sum
Calculating some observable, I obtained the following-like converges sum
$$
S = \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{p=0}^{\min(n,k)} \frac{x^n}{n!} \frac{y^k}{k!} F(p),
$$
where $F$ - some ...
- 121
0
votes
2
answers
119
views
How do I find the partial sum of the Maclaurin series for $e^x$?
In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$.
Wolfram | ...
- 3,515
1
vote
1
answer
151
views
Is it possible to express a power series with squared coefficients as a function of the series without squared coefficients?
Suppose I have two sums, $P(x)$ and $Q(x)$:
$$P(x)\equiv \sum_{n=0}^N a_n x^n$$
$$Q(x)\equiv \sum_{n=0}^N a_n^2 x^n$$
Is there a way to express $Q(x)$ as a function of $P(x)$?
Context: I have a ...
- 111
0
votes
0
answers
59
views
Interchange of differentiation and summation in infinite sums
Is it possible to interchange differentiation and summation for infinite but also uniformly convergent sums, like:
$\dfrac{d}{dx} \sin{x} = \dfrac{d}{dx}\sum_{n=0}^\infty (-1)^{n} \, \dfrac{x^{2n + 1}}...
- 181
0
votes
0
answers
67
views
Is it possible to rewrite this sum in terms of some power series?
Is it possible to rewrite this sum in terms of some power series, maybe some cosine power series?
$$\sum_{n=0}^{\infty} \dfrac{x^{2n}}{2^{2n}(n!)^2}$$
- 181
-1
votes
1
answer
63
views
Finding the nth sum of a series [closed]
I am to find the sum of a series that takes this format
$
\sum_{i=1}^{n}\frac{1}{i^\beta}
$
$
\beta
$
is a positive real number
How to approach the partial sum of the above series and can obtain its ...
- 3
0
votes
1
answer
59
views
Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Let $(a_i), (b_i)$ be two non-negative sequence.
Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Does it necessarily mean that ...
- 5,473
0
votes
1
answer
59
views
$\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$
I am stuck in finding the sum of $\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$.
The sum looks quite similar to a negative binomial sum but I can't really find the exact form. Can anyone help?...
- 243
6
votes
3
answers
325
views
Integral Representation of a Double Sum
Let us assume we know the value of $x$ and $y$. I'm trying to write the following double sum as an integral. I went through many pages and saw various methods but I'm completely lost with my problem.
$...
user977415
0
votes
1
answer
54
views
Question about this power series
I know that $\displaystyle\sum_{n=0}^{+\infty} \dfrac{(-1)^nx^{2n+1}}{2n+1}$ is the power series expansion for $f(x) = \tan^{-1}x$. The interval of convergence for this series is $(-1,1]$.
If I ...
- 1,830
4
votes
2
answers
237
views
General formula for the power sum $\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \mathbb{N}$
In my last question, it turns out to be solving the formula of $\sum_{k=1}^{n}k\omega^k$. I am curious if there is a geranal formula for the power sum:
$$\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \...
- 836
0
votes
0
answers
32
views
I don't know how to solve this summation of series
Here is the question
$\sum_{x=1}^{\infty} 2^{x(t-1)}$, where $t$ is a constant
Compare this summation with $Z^c$, where $Z \geq 0$, and specify for what values of $Z$ and $c$, such that $\sum_{x=1}^{\...
- 183
0
votes
1
answer
58
views
Can we reduce the 3-nested summation into 2-nest summation?
My problem started with the first case that I have :
$$
I_{2}=a^{k_1} (a+b)^{k_2}
$$
Where $k_1,k_2$ are real positive integers. Using the series expansion :
$$
(a+b)^n=\sum_{i_1=0}^{n} \binom{n}{i_1} ...
- 45
0
votes
0
answers
65
views
How to find the sum of a power series without knowing the actual power series
How do I find the sum of this series?
$$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} $$
The approach I wanted to use is to find a power series that can become this number series for a certain ...
- 11
0
votes
1
answer
71
views
Evaluate $\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$
Evaluate $$\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$$ using the fact that
$${\frac{n}{n^2-1}} = {\frac{1}{2(n-1)}} + {\frac{1}{2(n+1)}}$$
So far I have proven that the Radius of Convergence is 1 and ...
- 63
1
vote
2
answers
121
views
Upper bound for infinite sum
Let $x \in (0,1)$, I need to find an upper bound (as good as possible) for the series
$$\sum_{n=1}^{\infty}x^n n^k,$$
where $k$ is a natural number larger or equal than $2$, i.e., $k=2,3,4,\dots$.
My ...
- 35.9k
2
votes
1
answer
148
views
How to expand the product of Laguerre polynomials into a sum of series？
In the course of my research, I needed a formula and found it, but I can not understand the derivation process of the formula. How to extract the $t^n$ and get the $\theta(m-p)$ in the last step? Can ...
- 145