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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Formula for the sum of a geometric series

I'm using a book for my AS Level maths which says that: "The general rule for the sum of a geometric series is $$S_n = a\frac{r^n-1}{r-1}$$ or $$S_n= a\frac{1-r^n}{1-r}$$ " Why are there two formulas and when do I use each? Thanks so much, I'm…
Sami
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How to determine the limit of this sequence?

I was wondering how to determine the limit of $ (n^p - (\frac{n^2}{n+1})^p)_{n\in \mathbb{N}}$ with $p>0$, as $n \to \infty$? For example, when $p=1$, the sequence is $ (\frac{n}{n+1})_{n\in \mathbb{N}}$, so its limit is $1$. But I am not sure how…
Tim
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Series Proof $\sum_{k=1}^n (1/k) > \ln(n+1)$

Prove that $\sum_{k=1}^n (1/k) > \ln(n+1)$. I have been trying to do this for some time now, but I cannot figure it out. It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. Thanks So I know that…
Brian
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The sequence $\frac{2}{2-u_n}$ diverges

Let $(u_n)$ be a sequence defined with $u_{0}$ a real number such that $u_0 \notin \{0,1,2\}$ and $$u_{n+1} = \frac{2}{2-u_n}$$ Prove that $(u_n)$ diverges. I try to use the fact that this sequence fluctuates, having negatives values followed by…
Rono
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Convergence of $\sum \frac{a_n}{1+a_n}$ when $\sum a_n$ and $\sum a_n^2$ converges.

Suppose $a_n$ are real numbers and $\sum a_n$ and $\sum a_n^2$ converges. How would one go about showing that $\sum \frac{a_n}{1+a_n}$ converges? ($a_n \neq -1$ for every $n$)
John Smith
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Set of cluster points of a bounded sequence

Suppose that $\{\alpha_k\}$ is a bounded sequence of real numbers satisfying the condition $\displaystyle\lim_{k\rightarrow\infty}|\alpha_k-\alpha_{k+1}|=0$. Let $\displaystyle m = \varliminf_{k\rightarrow\infty}\alpha_k$ and $\displaystyle M =…
impartialmale
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Series Expansion at $n=\infty$ for $\frac{(2n-1)!!}{(2n)!!}$

Looking at this question, I asked Wolfram and got a Series Expansion at $n=\infty$ for $\displaystyle \frac{(2n-1)!!}{(2n)!!}$ like $\displaystyle \left({n^{-1/2}}…
draks ...
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How to evaluate the sum $\sum_{k=2}^{\infty}\log{(1-1/k^2)}$?

How to evaluate the sum $$\sum\limits_{k=2}^{\infty}\log{(1-1/k^2)}\;?$$
Hami
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How to solve the summation of series $a^{i}(x+i)$ where $i$ is from $1$ to $N$

I have the following series and I am unable to figure out which series it belongs to and how to solve it $a(x+1)+a^{2}(x+2)+…+a^{N}(x+N)$ Above series is a generalization of my actual series $\dfrac{1}{2}(x+1)+\dfrac{1}{4}(x+2) +\cdots…
user4131265
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Proving that the sum $\sum_{n=1}^{\infty }\frac{\sin^m(n\pi/2)}{n^2}=\frac{\pi^2}{8}$

Proving that the sum $$\sum_{n=1}^{\infty }\frac{\sin^m(n\pi/2)}{n^2}=\frac{\pi^2}{8}$$ When $m$=integer even number I know that the $\frac{\pi^2}{8}$ comes from $\sum_{n=0}^{\infty }\frac{1}{(2n+1)^2}$ and I know how to prove it but I don't…
user187581
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Problem in arithmetic progression

Three numbers $a,b,c$ are in arithmetic progression. How do we prove that $a^3 + 4b^3 + c^3 = 3b(a^2 + c^2)$? I need a proof that starts with LHS expression and arrives at RHS expression.
Thale
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To show that $\sum a_nb_n$ is absolutely convergent

Assume that $\sum a_n$ is convergent and $\sum b_n$ is absolutely convergent.To show that $\sum a_nb_n$ is absolutely convergent My try::Consider the sequence of partial sums of $\sum |a_nb_n|$ $S_n=|a_1b_1|+|a_2b_2|+......+|a_nb_n|$ Now…
Learnmore
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Find $\sum_{k_1+\cdots+k_n=m}\frac{1}{k_1!\cdots k_n!}$.

Problem: For fixed $m,n\in\mathbb{N}$, find $$\sum_{k_1+\cdots+k_n=m}\frac{1}{k_1!\cdots k_n!}$$ where the sum is over all integers $k_i\geq 0$ such that $k_1+\cdots+k_n=m$. I tried to come up with a series with the above coefficients, but I…
Orthogone
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General Term of a Sequence

What would be the best way in finding a general term $a_n, n>3$ for the recursive sequence? $$a_n = \dfrac{6a_{n-1}^2a_{n-3} -8 a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}$$ where $a_1 = 1 ; a_2 = 2 ; a_3 = 24$ ;
Afonso Henriques
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Help with evaluating this (likely telescopic) summation.

I am trying to solve this problem which is regarding evaluating summation: $$\sum_{k=1}^{\infty}\frac{6^{k}}{(3^{k}-2^{k})(3^{k+1}-2^{k+1})}$$ Points to note: It seems to be telescopic summation (very likely) and I am trying to create a telescopic…
Devarsh Ruparelia
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