All Questions
Tagged with summation power-series
362
questions
0
votes
1
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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 ...
- 268k
5
votes
1
answer
116
views
Studies about $\sum_{k=1}^{n} x^{\frac 1k}$
Are there any studies about this function?
$$f(x,n)=\sum_{k=1}^{n} x^{1/k}=x+x^{1/2}+x^{1/3}+x^{1/4}+\cdots +x^{1/n}$$
EDIT:
My first notes about it.
$f(1,n)=n$
$f'(1,n)=H_n$
$\int_0^1 \frac{f(x,...
- 268k
2
votes
1
answer
69
views
I need Help to evaluate :$\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 :
$$\int^{\frac{\pi}{2}}_0\cos^{2n+2}(x)dx=\int^{\frac{\pi}{2}}_0\sin^{2n+2}(x)...
- 268k
2
votes
1
answer
73
views
I need Help to evaluate series :$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$
I need Help to evaluate series :$$\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}}-...
- 3,348
28
votes
6
answers
4k
views
Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$
I am wondering if there exists any formula for the following power series :
$$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$
Is there any way to calculate the sum of above series (if ...
- 17.5k
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 ...
- 1,530
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:
$$\sum_{k=1}^...
- 268k
1
vote
2
answers
100
views
Find sum of power series
The problem is to find the sum of the power series:
$$\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)(n+1)}$$
My solution:
First to find where the sum exists (for which x):
Using D'Alembert's criterion for ...
- 2,428
0
votes
0
answers
62
views
Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]
I am looking for ways to simplify the sum
$$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$
The first thought I had approaching this was to use Hurwitz and/or Zeta ...
- 45
0
votes
0
answers
35
views
Variants of geometric sum formula
I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$
and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$.
(See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From Sum of ...
60
votes
11
answers
127k
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The idea behind the sum of powers of 2
I know that the sum of powers of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.
For example, sum of n numbers is $\...
- 61.5k
1
vote
1
answer
130
views
Borel Regularization of $\sum_{n=1}^\infty \ln(n)$ [closed]
I'm trying to solve the following taylor series
$$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$
so I can regularize the following sum
$$\sum_{n=1}^\infty \ln(n)$$
Using Borel Regularizaiton I can use ...
- 16.2k
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\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 $8.1$
- 1,961
5
votes
0
answers
102
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} \...
- 211k
4
votes
2
answers
241
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:
$$\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \...
- 482
0
votes
2
answers
119
views
How do I find the partial sum of the Maclaurin series for $e^x$?
In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$.
Wolfram | ...
- 2,269
1
vote
1
answer
99
views
Evaluate the sum of $1/n^6$ using Euler's method
I've just learned how Euler evaluated $1+1/2^2+1/3^2+...+1/n^2+...=\pi^2/6$ by comparing the coefficients of the series form and product form of $\sin(x)/x$.
The series form is
$$\dfrac{\sin(x)}{x}=1-\...
- 3,185
4
votes
1
answer
89
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.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 33.2k
2
votes
1
answer
251
views
What are the conditions for Ramanujan's Master Theorem to hold?
Ramanujan's Master Theorem states that if
$$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).$$
But there are obviously some conditions ...
- 5,337
0
votes
0
answers
23
views
Summation of the following form with non-integer n
I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
- 119
2
votes
3
answers
174
views
Sum to infinity series: $\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$
Consider the following series $$\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$$
I'm told to analytically find the sum to infinity and I have been given this as a clue.
$$\Sigma_{k=0}^\infty x^k = ...
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(...
- 268k
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 $p$. $A$, $B$,...
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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:
$...
- 50.3k
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} \...
- 20.7k
4
votes
2
answers
135
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)!}\...
- 109k
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
2
votes
1
answer
167
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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
$$\lim_{n\to+\infty}\left(...
- 268k
0
votes
1
answer
145
views
Solving a sum similar to geometric series
How do I solve the sum
$$\sum_{k=1}^y \left( 1-\frac{1}{\ln x} \right)^k \hspace{0.5cm} $$
for $x>0$ and $y$ a positive integer greater than one?
Despite resembling a geometric series, it does not ...
- 2,044
0
votes
1
answer
76
views
Prove the formula $1+r\cdot \cos(α)+r^{2}\cos(2α)+\cdots+r^{n}\cos(nα)=\dfrac{r^{n+2}\cos(nα)-r^{n+1}\cos[(n+1)α]-r\cosα+1}{r^{2}-2r\cdot \cos(α)+1}$
For $r,a\in\mathbb{R}:\; r^{2}-2r\cos{a}+1\neq 0$ prove the formula $$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 \...
- 34.4k
1
vote
1
answer
94
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 ...
- 109k
33
votes
3
answers
4k
views
Sum of Squares of Harmonic Numbers
Let $H_n$ be the $n^{th}$ harmonic number,
$$ H_n = \sum_{i=1}^{n} \frac{1}{i}
$$
Question: Calculate the following
$$\sum_{j=1}^{n} H_j^2.$$
I have attempted a generating function approach but ...
- 1,181
9
votes
1
answer
1k
views
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?
- 28.7k
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 ...
- 146k
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....
- 109k
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 ...
- 2,549
1
vote
0
answers
57
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
6
votes
2
answers
225
views
Does $ \sum_{i=1}^n n^{k_i} $ determines $(k_1,...,k_n)$?
Let $k_1,...,k_n\in\mathbb{N}$. Does the power sum
$$
\sum_{i=1}^n n^{k_i}
$$
uniquely determines the $n$-tuple $(k_1,...,k_n)$?
Remark: In the case $n=2$, this is true. However, when trying to ...
- 7,021
-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)} $ ?
- 268k
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
4
votes
6
answers
964
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 ...
- 99.8k
1
vote
1
answer
152
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 ...
- 3,251
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?
$$\sum_{n=0}^{\infty} \dfrac{x^{2n}}{2^{2n}(n!)^2}$$
- 33.2k
-1
votes
1
answer
64
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 ...
- 1,048
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 ...
- 18.6k
1
vote
4
answers
190
views
How do I express the sum $(1+k)+(1+k)^2+\ldots+(1+k)^N$ for $|k|\ll1$ as a series?
Wolfram Alpha provides the following exact solution
$$ \sum_{i=1}^N (1+k)^i = \frac{(1+k)\,((1+k)^N-1)}{k}.$$
I wish to solve for $N$ of the order of several thousand and $|k|$ very small (c. $10^{-12}...
- 7,103