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Results tagged with sequences-and-series
Search options answers only
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user 11919
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.
37
votes
Accepted
Identity for an infinite product
The identity is true. To see this, we consider
$$f(q):=q\prod_{k\geq 1}(1-q^k)^{24},\qquad |q|<1.$$
With this notation,
$$\frac{qf(q)}{f(q^2)}=\prod_{\substack{k\geq 1\\\text{$k$ odd}}}(1-q^k)^{24}\qq …
- 105k
4
votes
Accepted
Divergent summation
Define $x_i$ recursively as follows. Let $x_0:=1$, and for $i\geq 1$ let
$$x_i:=\left(\sum_{k=0}^{i-1} x_k \right)^2.$$
Then each term in the series equals $1$, hence the series diverges.
- 105k
6
votes
Accepted
Finding $\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0...
The right-hand sides of $(1)$ and $(2)$ are the same, because
$$\psi_0(1-x)=\psi_0(-x)-x^{-1}.$$
Hence we only need to show that the left-hand sides of $(1)$ and $(2)$ are also the same. That is, for …
- 105k
3
votes
How to prove that $ \sum_{k=1}^\infty \frac{\sin kx}{k^z} = \frac{1}{\Gamma(z)} \int_0^\inft...
Using the decomposition
$$\frac{e^t\sin x}{1–2e^t\cos x+e^{2t}}
=\frac{1}{2i}\left(\frac{1}{1-e^{t+ix}}-\frac{1}{1-e^{t-ix}}\right),$$
we obtain for $\Re(z)>1$ (using the Bose-Einstein integral) that
…
- 105k
12
votes
Is this formula expressing $\operatorname{erf} \left(\tfrac 1 {\sqrt 2}\right)$ as an infini...
WolframAlpha returns this result promptly, so it is probably known or straightforward to show. See here.
- 105k
12
votes
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3...
Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th …
- 105k
9
votes
Asymptotic for Ramanujan's $\tau$-function
No, $\tau(n)$ fluctuates wildly, and it cannot be described in simpler terms. It is "irreducible arithmetic data", and we just love that. Same for its absolute value.
- 105k
2
votes
Accepted
convergence for a series
This is not a research level question, but I feel like answering it. Simply, use the elementary estimate
$$\log(m!)\sim m\log m.$$
The symbol $\sim$ means that the ratio of the two sides tends to $1$. …
- 105k
2
votes
Accepted
proving inequality in Riemann zeta function
The inequality you want to prove is false. For example,
$$f_5(1/2,1)=0.6096\dots,$$
while
$$\lim_{k\to\infty}f_k(1/2,1)=0.6398\dots$$
- 105k
18
votes
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
It is straightforward to show that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. Indeed, if this equals a rational number $r\in\mathbb{Q}$, then $2\sin(2\pi r)=\sqrt{4/3}$. However, $2\sin(2\pi r)$ …
- 105k
8
votes
Accepted
Does the limit of $x_n$, defined by $x_{n+1}=1/(m+1-nx_n)$ exist?
The sequence $(x_n)$ does not have a limit. Let us assume, for a contradiction, that the limit exists.
The limit cannot be nonzero or $\pm\infty$, because then $|m+1-nx_n|\to\infty$ by the triangle in …
- 105k
6
votes
Surprisingly long closed form for simple series
The intuition becomes clear when we evaluate the series by bare hands.
For $r\in\{1,\dotsc,A\}$ we have
\begin{align*}\sum_{n=0}^\infty \frac{1}{A^n(An+r)}&=
\sum_{\substack{m\geq 1\\m\equiv r\pmod{A} …
- 105k
8
votes
Accepted
Difference between $n$-th and $(n-1)$-th composite numbers
Surely $f(n)$ is meant to be the indicator function of the range of the function $k\mapsto p(k)-k$, where $p(k)$ denotes the $k$-th prime number. With this supplemented definition, the conjecture is t …
- 105k
27
votes
Accepted
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Here is an elementary proof. We rewrite the series as
$$\frac{1}{4}\int_0^1\frac{1-x^4}{1-x^6}\,dx=\frac{1}{8}\int_0^1\frac{dx}{1-x+x^2}+\frac{1}{8}\int_0^1\frac{dx}{1+x+x^2}.$$
It is straightforward …
- 105k
10
votes
Accepted
About $\lim_{n\to +\infty} n\prod_{k=1}^{n-1}\cos^2(k)$
It follows from the irrationality of $\pi$ and Weyl's criterion that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the integers $k\in\{1,\dotsc,n-1 …
- 105k