All Questions
Tagged with summation power-series
362
questions
1
vote
5
answers
118
views
How do you call this fact about sum of powers of n-th unity root?
I often see identity
$$\sum_{k=0}^{n-1}e^{\tau ika/n} = \cases {n \quad \text{ if }n | a\\0\quad \text{ otherwise}}$$
in the context of generating functions. It allows to zero out all members of ...
- 1,481
3
votes
2
answers
2k
views
Geometric Series with coin tosses
Suppose you toss a coin and observe the sequence of $H$’s and $T$’s. Let $N$ denote
the number of tosses until you see “$TH$” for the first time. For example, for the
sequence $HTTTTHHTHT$, we needed $...
- 143
1
vote
2
answers
165
views
Is $\bigl(\sum {{x^n}\over{n!}} \bigr) \bigl(\sum {{y^n}\over{n!}} \bigr) = \bigl(\sum {{(x+y)^n}\over{n!}}\bigr)$ generalizable for series?
Before I had to do a proof demonstrating the properties of exponential multiplication using power series expansions: $$ e^xe^y=e^{x+y}, $$ and the easiest and quickest way I could think of doing this ...
- 165
1
vote
1
answer
105
views
Prove by induction that $\sum_{\varnothing\ne S\subseteq[n]}(\prod S)^{-1}=n$.
I'm having a hard time visualizing how to prove the following by induction:
For every positive integer $n$, let $[n]$ denote the set $\{1,\ldots,n\}$.
Let $A$ be a set. Use the notation $P(A)$ to ...
- 15
-1
votes
4
answers
66
views
Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?
Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?
I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has $\frac{t^{k+1}}{(k+1)!}$...
- 12.1k
1
vote
1
answer
89
views
Can this infinite summation be simplified?
I encountered the following infinite summation $$\sum_{k=0,k\neq m}^{\infty}\frac{x^k}{(k-m)k!},x>0,$$ can it be simplified? Thanks!
- 161
2
votes
1
answer
82
views
Series Solution to Differential Equation
Given the series $$1 + \sum_{k = 1}^{\infty} \frac{\beta(\beta - 1) ... (\beta - k + 1)}{k!} x^k$$ how can I find a differential equation for which this series is a solution?
I don't have any idea ...
- 861
2
votes
2
answers
93
views
Question on the sum $\sum_{n=1}^{\infty}\frac{x^n}{n} = -\ln(1-x)$
$f(x) = \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... = -\ln(1-x)$ for $|x| < 1$.
$f'(x) = \displaystyle\sum_{n=1}^{\infty}x^{n-1} = 1 + x + x^2 + x^3 +......
- 987
4
votes
1
answer
4k
views
Exponential series is cosh(x), how to show using summation?
I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n}}{(2n)!}
$$
I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that $$...
- 635
1
vote
1
answer
39
views
What is $s_3$ and $s_4$ for $x$
$\sum_{i=0}^n i^k = s_k(n)$,
$s_k$ polynomial from degree $k+1$
I have already shown for $s_2(x) = \frac{x(x+1)(2x+1)}6$
How from the sum and $s_2(x)$ can be shown for $s_3(x)$ and $s_4(x)$ ...
1
vote
4
answers
190
views
How do I express the sum $(1+k)+(1+k)^2+\ldots+(1+k)^N$ for $|k|\ll1$ as a series?
Wolfram Alpha provides the following exact solution
$$ \sum_{i=1}^N (1+k)^i = \frac{(1+k)\,((1+k)^N-1)}{k}.$$
I wish to solve for $N$ of the order of several thousand and $|k|$ very small (c. $10^{-12}...
- 981
0
votes
1
answer
2k
views
By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$
I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration.
Can anyone ...
user184036
0
votes
1
answer
54
views
Series for $(1-z)^{-\frac{1}{2}}$ and application?
How could I obtain
$$\sum_{k=0}^{\infty} {4k \choose 2k} \frac{z^{2k}}{2^{4k}}$$
from
$$\sum_{k=0}^{\infty} {2k \choose k} \frac{z^k}{2^{2k}}$$
which is $(1-z)^{-\frac{1}{2}}$. I can't manage to ...
- 205
6
votes
1
answer
130
views
Finding a bound for $\sum_{n=k}^l \frac{z^n}{n}$
For $z\in\mathbb{C}$ such that $|z|=1$ but $z\neq1$ and $0<k<l$, I'm trying to prove that:
$$\left|\sum_{n=k}^l \frac{z^n}{n}\right| \leq \frac{4}{k|1-z|}$$
It's more of a game that slowly ...
- 205
5
votes
1
answer
293
views
Solving 2nd order ODE with Frobenius method - problems with summation symbol
I'm trying to solve the ODE:
$$ y''(x) + \frac{2x}{(x-1)(2x-1)} y'(x) - \frac{2}{(x-1)(2x-1)} y(x) = 0 $$
I'm trying to find a solution by the Frobenius method, expanding a power series of the ...
- 81