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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Find sum of alternating series $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {n^2}$.

Find sum of series $$\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {n^2}$$. I know the series converge absolutely so it is clearly convergent and in the absolute case the sum is $\pi^2/6$. However, I can't seem to find the sum in this case ? Also the…
user141901
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How to sum this series to infinity: $\sum_{n=0}^{\infty} \frac1{2^{2^n}}$

How to sum the series: $$\sum _{ n=0 }^{ n=\infty }{ \frac { 1 }{ { 2 }^{ { 2 }^{ n } } } }$$ PS: Just a hint would suffice.
Tom Lynd
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What can be said about a series with nonzero terms, whose sum is zero?

What can be said about an infinite series $\sum^{\infty}_{n=1} a_n$ with $a_n \neq 0$ for all $n$, whose sum is zero ? Does such a series exist ? If yes, can you give an example ?
the8thone
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Transcendentals that sum to rationals

Is there any sequence $a_n$ of transcendental numbers, such that $ma_i\neq na_j$ for all integers m,n,i,j and any partial sum of $a_k$ is transcendental, but the total sum is rational ?
TROLLHUNTER
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Simple demonstration for $\lim_{n\to\infty}(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}) = \frac{1}{e}$

What is the simple demonstration with elementary means for Lalescu Sequence: $$\lim_{n\to\infty}(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}) = \frac{1}{e}?$$ (Traian Lalescu-romanian mathematician (1882-1929))
medicu
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How to prove $\sum\limits_{n=p+1}^\infty \frac1{n^2-p^2} = \frac1{2p} (1+\frac12 + \cdots + \frac1{2p})$ for all $p \in \mathbb N$?

For all $p \in \mathbb N$, I want to prove $$\sum_{n=p+1}^\infty \frac1{n^2-p^2} = \frac1{2p} \left(1+\frac12 + \cdots + \frac1{2p} \right).$$ Up to now, I've approached the problem using induction and/or partial fractions. When I use induction, I…
Huy
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Show that $\sum_n \frac{1}{a_n}\lt90$

Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all the positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $$\sum_n \frac{1}{a_n}\lt90$$ I would try to…
Guy
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Evaluate $\sum_{n=1}^\infty \frac{n}{2^n}$

This is a homework question; I'm supposed to use power series to find the following sum: $$\sum_{n=1}^\infty \frac{n}{2^n}$$ I took the geometric series $$\frac{1}{1-x}=\sum_{n=0}^\infty {x^n}$$ and differentiated and multiplied both sides by x to…
Zach
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Convergence of $\frac{\sin(g(n))}{f(n)}$

What do we require of $g(n)$, if for every positive strictly increasing unbounded $f(n)$, this sum converges? $$\sum_{n=1}^{\infty} \frac{\sin(g(n))}{f(n)} .$$ Does it converge for $g(n)=n$?
TROLLHUNTER
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Manipulating harmonic series

Given the harmonic series has the summation $$ \sum_{k=1}^n\frac1k=\ln|n| + O(1)$$ How do we show that: $$ \sum_{k=1}^n\frac{1}{2k-1}=\ln\left|\sqrt{n}\right| + O(1)$$
oliland
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How find this $a_{n},b_{n}$

let sequence $\{a_{n}\},\{b_{n}\}$ such $$a_{1}=1,b_{1}=3$$ and $$\begin{cases} a_{n+1}=2+\dfrac{27a_{n}}{9a^2_{n}+4b^2_{n}}\\ b_{n+1}=\dfrac{27b_{n}}{9a^2_{n}+4b^2_{n}} \end{cases}$$ Find $a_{n},b_{n}$ My idea:…
user94270
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Which series converges question

Which of the following statements are true. Given $S_1, S_2$, where $S_1:$ A series $$\sum_{n=0}^{\infty}a_n$$ converges if for a given $\epsilon\gt0$ there exists $N_o \in N$ such that $|a_{n+1}-a_{n}|\lt \epsilon$ for all $n\ge N_o$. $S_2:$ A…
tattwamasi amrutam
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Why can infinite series be summed different ways to get different results?

$$S = 1 - \frac12 + \frac13 - \frac14 + \frac15 - \frac16 + \frac17 - \frac18 + \frac19 - \frac1{10} + \frac1{11} - \frac1{12}\ldots\text{(to infinity)}$$ Rearranged, this series looks like: $$S = \left(1 - \frac12\right) - \left(\frac14\right) +…
Niobius
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A possible theorem

So i was playing around with members of a random power set, and i came to a revelation(at least to me it was). Say $A=\{1,2,3\}$ then for arbitrary $k,n\in Z^+$, $n=|A|$ and…
pkjag
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What does $\frac{1}{n}$ converge to?

What does the sequence of $n=1$ to infinity converge to for $\dfrac{1}{n}$ and how do I prove this? I understand that as $n$ gets bigger, the fraction gets smaller, but how do I find the exact value it converges to?
Lauren
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