All Questions
Tagged with summation power-series
362
questions
0
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answers
59
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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
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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
9
votes
3
answers
13k
views
Formula for finite power series
Are there any formula for result of following power series?
$$0\leq q\leq 1$$
$$
\sum_{n=a}^b q^n
$$
- 1,536
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?...
- 141k
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.
$...
- 109k
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 ...
- 58.3k
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 :
$$
(a+b)^n=\sum_{i_1=0}^{n} \binom{n}{i_1} ...
- 181k
0
votes
0
answers
65
views
How to find the sum of a power series without knowing the actual power series
How do I find the sum of this series?
$$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} $$
The approach I wanted to use is to find a power series that can become this number series for a certain ...
- 151k
0
votes
1
answer
71
views
Evaluate $\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$
Evaluate $$\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)}}$$
So far I have proven that the Radius of Convergence is 1 and ...
- 330
2
votes
1
answer
151
views
How to expand the product of Laguerre polynomials into a sum of series？
In the course of my research, I needed a formula and found it, but I can not understand the derivation process of the formula. How to extract the $t^n$ and get the $\theta(m-p)$ in the last step? Can ...
- 109k
1
vote
2
answers
121
views
Upper bound for infinite sum
Let $x \in (0,1)$, I need to find an upper bound (as good as possible) for the series
$$\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 ...
- 170k