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Results tagged with sequences-and-series
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user 101078
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.
2
votes
$\sum\limits_{n=1}^{\infty}(-1)^n (\frac{a_{n+2}}{a_{n}}-1)$ converges. Does $\sum\limits_{n...
The following sequence provides a counterexample and satisfies assumptions 1,2 and 3:
$$a_{2n+1}=(n+1)(n+2), a_{2n}=n(n+1)+1=a_{2n-1}+1.$$
To check this, let
$$b_n=(-1)^n(\frac{a_{n+2}}{a_n}-1), c_n …
- 7,193
20
votes
Accepted
Under a condition, $\frac{1}{b } = \sum_{n=1}^{\infty}\frac{1}{a_{n}}$ will never happen
Actually, this happens for all natural $b$. Notice that
$$
\frac{1}{b}=\frac{1}{b+1}+\frac{1}{b^2+b}
$$
and iterate this identity. You will get
$$
\frac{1}{b}=\frac{1}{b+1}+\frac{1}{b^2+b+1}+\frac{1}{ …
- 7,193
6
votes
Accepted
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
Let $a$ be the base of logarithm here. Consider the function $$f(x)=\frac{(x+1/2)\ln x-x}{\ln a}.$$
As noted in the question, it is enough to prove that the sequence $f(n)$ is dense modulo $1$. In fa …
- 7,193
1
vote
Bounds for the sequence $a(n,A)=n*a(\lfloor (1-A)n \rfloor,A)$
Define the sequence $n_k$ via the recursive formula $n_0=n$ and $n_{k+1}=\lfloor(1-A)n_k \rfloor$. As $0<A<1$, the sequence is decreasing. Let $M$ be the largest solution of inequality $n_M\geq 1$. Fi …
- 7,193
22
votes
Accepted
The constant $e$ represented by an infinite series
Your sum actually equals $\frac{\pi\sqrt{3}}{2}$, so it's more like $\pi$ all over again, not $e$. To see this, note first that by definition $\mathrm{sgn}_2$ is multiplicative, hence
$$
A=\sum_{n=1}^ …
- 7,193
1
vote
Understand the properties of this function
There is a meromorphic continuation of $g(t)$ with only pole of order $2$ at $t=0$. To prove this, lets consider the Mellin transform $G(s)$ of $g(t)-1$. It is easy to see that $g(t)-1$ decreases expo …
- 7,193