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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Sum $\frac {1}{9} + \frac {1}{18} +\frac {1}{30} +\frac {1}{45} +\frac {1}{63} \ldots$

What is the sum of the following series: $\frac {1}{9} + \frac {1}{18} +\frac {1}{30} +\frac {1}{45} +\frac {1}{63} \ldots$ I tried to do it through telescope method but it didn't work.
Shivam Kumar
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By knowing, that $a_{n+1}=(n+3)a_n$, how can I find $a_n$?

$$a_1=1$$ $$a_{n+1}=(n+3)a_n$$ How can I get to the answer of this, which is: $$a_n=\frac{(n+2)!}{6}$$
Niki Bregvadze
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Prove that, if $\sum\limits_na_n$ converges, then $\sum\limits_n \frac{a_n}{a_{n+1}}$ diverges

Prove that, if $\sum\limits_na_n$ converges, then $\sum\limits_n \frac{a_n}{a_{n+1}}$ diverges. I believe it's true and even seems obvious, but once I tried to prove it I found it's not that trivial and I can't prove it rigorously.
Kay K.
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The sum of series $1\cdot 3\cdot 2^2+2\cdot 4\cdot 3^2+3\cdot 5\cdot 4^2+\cdots \cdots n$ terms

The sum of series $1\cdot 3\cdot 2^2+2\cdot 4\cdot 3^2+3\cdot 5\cdot 4^2+\cdots \cdots n$ terms i have calculate $a_{k} = k(k+2)(k+1)^2$ so $\displaystyle \sum^{n}_{k=1}a_{k} = \sum^{n}_{k=1}k(k+1)^2(k+2)$ i wan,t be able go further, could some…
DXT
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If $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ then $\sum_{n=1}^{\infty}(a_{n}-a_{n+2})$ converges

If $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ then $\sum_{n=1}^{\infty}(a_{n}-a_{n+2})$ converges Using the fact that $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ I showed that $a_{n}$ is Cauchy hence $a_{n}\to L$. If we let $b_{n}=a_{n}-a_{n+2}$…
user6751
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Prove that: $\tan^{-1}\left({1\over nx}\right)=\sum_{i=n}^{\infty}\tan^{-1}\left({1\over i(i+1)x+x^{-1}}\right)$

Show that $$\tan^{-1}\left({1\over nx}\right)=\sum_{i=n}^{\infty}\tan^{-1}\left({1\over i(i+1)x+x^{-1}}\right)\tag1$$ $(n,x)$ $\ge1$; x is any real numbers My try: Setting $(x,n)=1$ We have $$\sum_{k=1}^{\infty}\tan^{-1}\left({1\over…
gymbvghjkgkjkhgfkl
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How do I prove that a conditionally convergent series can diverge to infinity under a suitable rearrangement?

I can prove a conditionally convergent series can be made to converge to any real number ( using the idea of adding just enough positive terms to get to the number then adding just enough negative terms etc) but im not sure how to show it can be…
Sunny Sood
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Evaluating sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^a}\sum_{m=1}^{n}\frac{1}{m^b}$

How to tackle infinite sums of the type $$S(a,b)=\sum_{n=1}^{\infty}\frac{1}{n^a}\sum_{m=1}^{n}\frac{1}{m^b}$$ where $a$ and $b$ are natural numbers. Zeta function is involved, for example $S(3,2)=3\zeta(2)\zeta(3)-\frac{9}{2}\zeta(5)$, but so far I…
Stoyan Apostolov
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Monotonically increasing Riemann sums for monmials

For an integer $r\ge1$, consider the sequence $(a_n)_{n\ge2}$ defined by $$a_n=\frac{1}{n^{r+1}}\sum_{k=1}^{n-1}k^r$$ It is easy to prove, for $r=1,2,3$, that this sequence in increasing using the known closed form of the sum. But is it true in…
Omran Kouba
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Rearrangement of sigma notations

I'm told that the following is true when $m,n\in\Bbb Z$: $$\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n} \alpha_{ij}=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m} \alpha _{ij}$$ When $m,n\to\infty$, If both sides above have the same divergence or…
Michael Li
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Finding the sum of the components of a series.

I was reading sequence and series and there I saw the formula for the sum of the squares of the natural numbers ($1^2+2^2+3^2+\cdots+n^2$) so I just incurred a doubt about the sum obtained when these squared numbers are inversed i.e…
Mayank Mittal
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Show convergence of $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} c_n$ implies the convergence of $\sum_{n=0}^{\infty} b_n$

Let $(a_n), (b_n), (c_n)$ be sequences of real numbers such that $\forall n \in \mathbb{N} \ \lvert a_n-b_n \rvert \leq c_n$. I want to show that if the series $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} c_n$ converge, the series…
Zelazny
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Proof of the One-Day Theorem

In his paper John Conway has put forward some theorems describing the behaviour of the look-and-say sequence. The very first theorem, the One-Day Theorem, states that no one-day-old string contains substrings of the following types: $•$ $yxzx$ $•$…
Daphne
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How to show $\left(1+\frac{1}{3}x+\frac{1}{24}x^2+\frac{1}{300}x^3+...\right)^2 = 1 +\frac{2}{3}x+\frac{7}{36}x^2+\frac{1}{30}x^3+...$?

I tried to find the right handside of the equation by manipulating the series but I failed at getting the right handside of it. $$\left(1+\frac{1}{3}x+\frac{1}{24}x^2+\frac{1}{300}x^3+...\right)^2 = 1…
FreeMind
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Why does the geometric series formula intuitively work?

I understand the proof for the geometric series formula, but I don't understand how the formula, $S_n = a_1\frac{(r^n -1)}{(r-1)}$ actually relates to the sum of all the terms. What operations are taking place in the formula to give the sum.
Tom Brooks
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