Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
65,955
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Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums.
...
- 55.8k
144
votes
1
answer
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Identification of a curious function
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},
$$
where $...
- 57.3k
141
votes
36
answers
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 1,585
135
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7
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Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=...
123
votes
18
answers
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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
- 55.8k
122
votes
4
answers
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Motivation for Ramanujan's mysterious $\pi$ formula
The following formula for $\pi$ was discovered by Ramanujan:
$$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$
Does anyone know how it works, or ...
- 19.1k
117
votes
1
answer
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Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$
Is there a way to assess the convergence of the following series?
$$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$
From numerical estimations it seems to be convergent but I don't know how to prove it.
- 1,859
115
votes
25
answers
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Can an infinite sum of irrational numbers be rational?
Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear ...
- 1,841
115
votes
5
answers
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How to sum this series for $\pi/2$ directly?
The sum of the series
$$
\frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1}
$$
can be derived by accelerating the Gregory Series
$$
\frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2}
...
- 348k
113
votes
16
answers
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If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum_{n=1}^\infty a_n<\infty$, then $\lim\limits_{n\to\infty}{n a_n} = 0$
I'm studying for qualifying exams and ran into this problem.
Show that if $\{a_n\}$ is a nonincreasing sequence of positive real
numbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \...
- 4,666
110
votes
6
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Why is a geometric progression called so? [duplicate]
Just curious about why geometric progression is called so. Is it related to geometry?
- 1,401
108
votes
3
answers
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How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$
Nice problem:
Let $a_{0}=1$ and
$$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that
$$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\...
- 93.6k
104
votes
13
answers
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Limit of sequence in which each term is defined by the average of preceding two terms
We have a sequence of numbers $x_n$ determined by the equality
$$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$
The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $...
- 2,277
103
votes
4
answers
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A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
As the title says, I would like to launch a community project for
proving that the series $$\sum_{n\geq 1}\frac{\sin(2^n)}{n}$$ is
convergent.
An extensive list of considerations follows. The ...
- 355k
101
votes
1
answer
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Arithmetic-geometric mean of 3 numbers
The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows:
$$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,...
- 47.3k