Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Sequences and series are often considered as a main part of part of calculus, in addition to limits, continuity, differentiation and integration.

A sequence is an enumerated collection of objects in which repetitions are allowed. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous; harmonic progressions, which is formed by taking the reciprocal of each term of an arithmetic progression; logarithmic progression, which is formed from a series whose progression getting smaller; and geometric progressions, where the next term is a constant multiplied by the previous term.

A sequence can be given by a direct formula (e.g. $a_n = 2^n + 3$), or by a recurrence relation. In a recurrence relation, the relation between the next term and the earlier terms is given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n, n \geq 0.$ Together with the initial terms $F_0=0$ and $F_1=1$, this recurrence relation defines the famous Fibonacci sequence.

A series is formed by summing a sequence. A typical question is: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is it convergent? Several tests, such as the ratio test, the root test, the limit comparison test, the integral test, etc. can help to answer these questions.

Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting these questions at Puzzling Stack Exchange instead.

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A formula of Ramanujan for $\cot\sqrt {w\alpha} \coth\sqrt{w\beta} $

While trying to answer this question I stumbled on a paper by Bruce C. Berndt which contains the following formula by Ramanujan $$\frac{\pi}{2}\cot\sqrt{w\alpha}\coth\sqrt{w\beta}=\frac{1}{2w}+\sum_{m=1}^{\infty} \left(\frac{m\alpha\coth m\alpha}…
Paramanand Singh
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Necessary condition of changing signs of a divergent series $\sum_{n=1}^{\infty}p_{n}$ to make it convergent,$p_{n}$ decreases and tends to $0$.

Let $p_{n}$ decreases and tends to $0$ while $\sum_{n=1}^{\infty}p_{n}$ is divergent. We choose $\varepsilon_{n}=\pm 1$ to make $\sum_{n=1}^{\infty}\varepsilon_{n}p_{n}$ convergent. I want to prove…
Tree23
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Infinite series - anecdote about John von Neumann

There are countless of anecdotes about John von Neumann and his dazzling intellect, one of which is the following: "Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the…
OmnipresentAbsence
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Evaluating series with factorial denominator (sanity check).

Is my approach to evaluating this series correct? $$\sum_{n=1}^\infty \frac{n}{(n+1)!}$$ Has partial sum equivalent to: $$S_m = \sum_{n=1}^m \frac{n}{(n+1)!} = \sum_{j=2}^{m+1} \frac{j-1}{j!} = \sum_{j=2}^{m+1} \frac{1}{(j-1)!} - \sum_{j=2}^{m+1}…
conjectures
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$\sum \frac{1}{f(k)}$ converges iff $\sum \frac{f^{-1}(k)}{k^2}$ converges

Let $f$ be a strictly increasing positive continuous function defined on $[1,\infty)$ with limit $\infty$ as $x$ goes towards $\infty$. Then $\sum_{k=0}^{\infty} \frac{1}{f(k)}$ converges if and only if $\sum_{k=0}^{\infty} \frac{f^{-1}(k)}{k^2}$…
Johan
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$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$ by integration or differentiation

There is an infinite sum given: $$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$$ It should be solved using integration, derivation or both. I think using power series can help but I don't know how to finish the calculation. Any help will be appreciated!
Hendrra
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Evaluation of $\sum _{n=1}^{\infty} \tan^{-1} \frac{2}{n^2+n+4}$

Find the following sum $$S= \sum _{n=1}^{\infty} \tan^{-1} \frac{2}{n^2+n+4}$$ I am not able to make it telescopic series. Could someone help me with this?
Mathematics
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Find sum of infinite anharmonic(?) series

I need help with this: $$ \frac{1}{1\cdot2\cdot3\cdot4}+\frac{1}{2\cdot3\cdot4\cdot5}+\frac{1}{3\cdot4\cdot5\cdot6}\dots $$ I don't know how to count sum of this series. It is similar to standard anharmonic series so it should have similar solution.…
SekstusEmpiryk
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Infinite sum involving number of solutions to $k=i^2+j^2$

I want to calculate the following sum: $$ S=\sum_{k=1}^\infty (-1)^{k-1}\frac{r_2(k)}{k} $$ Where $r_2(k)$ is the number of ways to write $k$ in the form $i^2+j^2$ where $i,j\in\mathbb Z$. I was able to transform it into: $$ \frac S 4…
Redundant Aunt
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Definition of a geometric sequence

Is the sequence $0, 0, 0, 0 ...$ geometric? If so how would you define it? In order to define a geometric sequence you need the first term, and the ratio of terms. In this case you could have: $a = 0$ $r = k$ for some $k \in \mathbb{R}$ Is this…
talfred
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Is Knopp's "Theory and Application of Infinite Series" out of date?

Is Knopp's Theory and Application of Infinite Series out of date? It's looks terrific to me, but the Dover edition I bought new maybe a year ago: http://preview.tinyurl.com/2eprqps seems to be the same as an edition published in 1951 and may go back…
NotSuper
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$\sum_{n=1}^{\infty}\frac{n^2}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}$is also convergent?

Let sequence $a_{n}>0$, $n\in N^{+}$, and such $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{a_{n}}$ convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{n^2}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}$$ is also convergent? Jack A related result: maybe I guess this…
New Zealand
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Formula for a periodic sequence of 1s and -1s with period 5

I've been playing with periodic sequences of 1s and -1s lately. This is what I came up with: \begin{eqnarray*} -(-1)^n& = &1, -1, 1, -1,\ldots\quad\textrm{(Period 2)}\\ \left(-(-1)^n\right)^{\frac{n+2}{2}} & = &1, 1, 1, -1, 1, 1, 1,…
Mikulas
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Calculate $\sum_{n=1}^{\infty}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4})$

I am trying to calculate the following series: $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)}$$ and I managed to reduce it to this term $$\sum_{n=1}^{\infty}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4})$$ And here I am stuck. I tried writing down a few…
Oria Gruber
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Binomial Expansion, Taylor Series, and Power Series Connection

1) Is there a reason why the binomial expansion of $(a+x)^n$ is the same as a Taylor series approximation of $(a+x)^n$ centered at zero? 2) The binomial expansion of $(a+x)^n$ is $a^n + na^{n-1}x + \frac{n(n-1)}{2!}a^{n-2}x^2 +$.... If the…
DWade64
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