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Results tagged with sequences-and-series
Search options answers only
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user 11260
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
1
vote
Accepted
Sum of series $\sum_{i=1}^n i^{-\alpha}$
$$\sum_{i=1}^n i^{-\alpha}=H_{n,\alpha}$$
the generalized Harmonic number. For $\alpha>1$ one has the limit
$$\lim_{n\rightarrow\infty}H_{n,\alpha}=\zeta(\alpha),$$
the Riemann zeta function. The larg …
- 188k
8
votes
Newton series and Fourier transform - is there an analogy?
to give some "exact math", expanding on my comment, here is the Poisson-Mellin-Newton cycle:
from $g(k)$ to $f_n$ via a Newton series: $f_n=\sum_{k=0}^n {n\choose k}(-1)^k g(k)$
from $f_n$ to $f(t)$ …
- 188k
4
votes
Accepted
Do Volterra series have a region of convergence, as do Taylor series?
The convergence issue of a Volterra series, basically a Taylor series for functionals, is summarized as follows by Scholarpedia:
Due to its power series character, the convergence of an infinite
…
- 188k
10
votes
Value of $c$ such that $\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\l...
UPDATE
I tried to evaluate the sum numerically for large $n$ and what I find does not support the conclusion I give below, that the large-$n$ limit equals 1 independent of $c$. Here is a plot for $c= …
- 188k
5
votes
Accepted
Closed form expression for this infinite series?
No closed form expression in terms of elementary functions, but if you are satisfied with special functions (incomplete Gamma function $\Gamma$ and exponential integral Ei), then
$$\displaystyle\sum_{ …
- 188k
1
vote
On double series involving Gregory coefficients and quotients of particular values of the ga...
The result in the OP
$$\sum_{n=1}^\infty \sum_{k=1}^\infty\frac{G_n}{k}\frac{\Gamma(k+\frac{3}{2})\Gamma(n+\frac{1}{2})}{(n+k+1)!}=\frac{\pi(1-8\log 2)}{8}=-1.78489,$$
does not seem to agree with a n …
- 188k
6
votes
What are some interesting relationships between pi and phi?
Q: Is there any way to use the golden ratio $\Phi$ to define $\pi$ ?
as proven by John Baez.
- 188k
13
votes
Evaluating an infinite sum related to $\sinh$
this is not a proof – GH has given that – but I just want to note four more general series of this type listed in Andreas Dieckmann's extensive collection:
the OP's sum is the fourth series (or the …
- 188k
3
votes
Reference request for a certain exponential series
Q: Is there a reasonable alternative expression for the sum
$$\sum_{d=-\infty}^\infty e^{-t^d}t^{kd},\;\;0<t<1,\;\; k\in\mathbb{N}.$$
A: I note the integral expression
$$\int_{-\infty}^\infty e^{-t^x} …
- 188k
8
votes
Accepted
Special values of $\sum_{n\ge1}z^n/(n^2\binom{2n}{n})$
I interpret the request of the OP for an "explicit" evaluation of the series as a request for a "closed-form" expression, which exists (it seems to go back to Euler, here are several proofs):
$$f(z)=\ …
- 188k
2
votes
Accepted
Solving recurrence relation for symmetric Toeplitz matrices determinant
$$T_n=\frac{1}{2^{n+1}} \left(c_-^n+c_+^n+\left(c_+^n-c_-^n\right)\frac{c_++c_- }{c_+-c_-}\right),\;\;\text{with}\;\; c_\pm=a\pm\sqrt{a^2-4 b^2}.$$
- 188k
12
votes
Accepted
Closed form of an infinite series
Q: Does the following infinite series have a closed form?
It does, according to Mathematica:
$$\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin\left(\frac …
- 188k
15
votes
Accepted
Closed form for $\sum_{n=0}^\infty \frac1{2^{2^n}}$?
If you allow for a named number to be a closed form representation, the answer is "yes".
$\sum_{n=0}^\infty (1/2)^{2^n}$ is known as the Kempner number [1], a transcendental number [2].
More generally …
- 188k
3
votes
Accepted
Does any such family of functions exist?
Here is a real solution for $k=2$: Take $f_1(x)=x$ and
$$f_2(x)=\arccos\left( \sqrt{\tfrac{3}{4}- \cos ^4(x/2)} \operatorname{cotan} (x/2)-\tfrac{1}{2}\cos x\right).$$
Then
$$\cos f_1(x)+\cos f_2(x)=\ …
- 188k
1
vote
Sum of an infinite series
A closed-form answer is not forthcoming, but for $c\lesssim 1$ the small-$c$ approximation is quite accurate:
$$S_0=\sum_{n=1}^\infty \frac{x^n}{n!}(1+c/n)=e^x-1+c \bigl[\text{Chi}(x)+\text{Shi}(x)-\l …
- 188k