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.

65378 questions
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Series problem! Can someone give me a counterexample?

Suppose positive series $\sum a_n<+\infty$, does this implies that $$\lim_{n\to\infty}na_n=0 .$$
Riemann
  • 7,203
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Value of $\sum_{n=1}^\infty\sin^{2k+1}(n)/n$

Prove or disprove: for any positive integer $k$, $$\sum_{n=1}^\infty\frac{\sin^{2k+1}(n)}{n}=\frac{1}{3}\left(\frac{3}{4}\right)^k\pi$$ This is a conjecture, which I haven't found anywhere else. I've checked it using Wolfram Alpha for small values…
Joe Corliss
  • 101
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What do these numbers signify?

During a visit in Nicosia I came across a strange sequence (which was situated in a rather bizarre place). What is the connection between the numbers or what do they signify?
Yair
  • 501
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Ramanujan's sum related to $\tan^{-1}(e^{-\pi x/2})$

While reading Ramanujan's Collected Papers I came across a nice formula which he mentions without proof $$\tan^{-1}(e^{-\pi x/2}) = \frac{\pi}{4} - \left(\tan^{-1}\frac{x}{1} - \tan^{-1}\frac{x}{3} + \tan^{-1}\frac{x}{5} - \cdots\right)$$ where $x$…
Paramanand Singh
  • 87,309
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Summing ${\frac{1}{n^2}}$ over subsets of $N$.

Are there $2$ subsets, say, $A$ and $B$, of the naturals such that $$\sum_{n\in A} f(n) = \sum_{n\in B} f(n)$$ where $f(n)={\frac{1}{n^2}}$? If $f(n)=\frac{1}{n}$ then there are many counterexamples, which is probably a consequence of the…
user414
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What is a necessary and sufficient condition for a Taylor series to exist?

I am trying to decide when a function can be written as a Taylor series. I think it exists if the following condition is met: For a Taylor series of $f(x)$ about the point $a$ In the region $R$ containing both $x$ and $a$, the function $f(x)$ is…
Quantum spaghettification
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Proofs for an equality

I was working on a little problem and came up with a nice little equality which I am not sure if it is well-known (or) easy to prove (It might end up to be a very trivial one!). I am curious about other ways to prove the equality and hence I thought…
user17762
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Ramanujan's formula for $\sum_{n = 0}^{\infty}p(7n + 5)q^{n}$

Analyzing the function $$f(-q) = (1 - q)(1 - q^{2})(1 - q^{3})\cdots$$ by replacing $q$ with $q^{1/5}$, Ramanujan is able to calculate the sum $$\sum_{n = 0}^{\infty}p(n)q^{n/5} = \frac{1}{f(-q^{1/5})}$$ where $p(n)$ is the number of partitions of…
Paramanand Singh
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does $\sum_{n=1}^{\infty} \frac{1}{\zeta(1+\frac{1}{n})}$ diverge or converge?

I'm asking because numerical tests seem to give nonsensical answers, and I thought I would check if there was an analytic way of checking for divergence, but I couldn't think of one offhand.
graveolensa
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Divergence of $\sum\limits_{k=1}^n \frac{a_n}{S_n}$ where $S_n = \sum\limits_{k=1}^n a_k$, when $S_n$ diverges

Let $a_n$ be a positive sequence such that $S_n = \sum\limits_{k=1}^n a_k$ diverges. I'm trying to prove $\sum\limits_{k=1}^n \frac{a_n}{S_n}$ diverges. I tried summation by parts, limit comparison and Stolz theorem in many different…
user136640
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Calculating squared reciprocals of arithmetic series

Let $n>0$ be an integer. Is it possible to calculate the value of the sum $$1+\frac1{(1+n)^2}+\frac1{(1+2n)^2}+\ldots$$?
Kunal
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How to compute infinite series $\sum_{n=0}^{\infty} ne^{-n}$

I'm trying to compute the infinite series $\sum_{n=0}^{\infty} ne^{-n}$. I know the answer is $e/(e-1)^2$, but I don't understand how to find this result. Thanks for the help!
koobtseej
  • 595
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Evaluate the series $\sum_{k\geq 1} \frac{1}{2^k k^2}$

How to prove that $$\sum_{k\geq 1} \frac{1}{2^k k^2}=\frac{\pi^2}{12}-\frac{1}{2}\log(2)^2$$ without using the well-known $\operatorname{Li}_2\left( \frac{1}{2} \right)$ ? Edited : Thanks for L.F , but I should have made it clear that I want an…
Zaid Alyafeai
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Intruiging Symmetric harmonic sum $\sum_{n\geq 1} \frac{H^{(k)}_n}{n^k}\, = \frac{\zeta{(2k)}+\zeta^{2}(k)}{2}$

I proved the following equation $$\sum_{n\geq 1} \frac{H^{(k)}_n}{n^k}\, = \frac{\zeta{(2k)}+\zeta^{2}(k)}{2}$$ We define $$H^{(k)}_n=\sum_{m= 1}^n \frac{1}{m^k}$$ I am looking forward to seeing what approaches would you use .
Zaid Alyafeai
  • 14,343
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Evaluating $\frac12+\frac14+\frac18+\frac1{10}+\frac1{20}+\frac1{22}+\frac1{44}+\frac1{46}+\frac1{92}+\cdots$

How can I find the value of this series? $$\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{10} + \dfrac{1}{20} + \dfrac{1}{22} + \dfrac{1}{44} + \dfrac{1}{46} + \dfrac{1}{92}+\cdots$$ The pattern is that the denominator increases by $2$…
İsmim Yok
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