All Questions
Tagged with summation power-series
362
questions
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
$$\lim_{n\to+\infty}\left(...
- 268k
0
votes
1
answer
145
views
Solving a sum similar to geometric series
How do I solve the sum
$$\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