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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Epsilon-delta proof for a sum

Consider $\displaystyle \begin{array}{ccccc} f & : & \mathbb R^+ & \to & \mathbb R \\ & & x & \mapsto & \frac{x}{\sqrt{1+x}} \\ \end{array}$ Prove that $\displaystyle \sum_{k=1}^n f\left(\frac{k}{n^2}\right)\to \frac{1}2$ The main trouble here is…
Gabriel Romon
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$\sum_{n=1}^{\infty} (a_n+b_n)=\sum_{n=1}^{\infty} a_n + \sum _{n=1}^{\infty} b_n$?

It is true to say that $\sum_{n=1}^{\infty} (a_n+b_n)=\sum_{n=1}^{\infty} a_n + \sum _{n=1}^{\infty} b_n$? It seems false but i can't find any counterexample..
user2637293
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$\sum_{k=1}^{\infty} \frac{a_1 a_2 \cdots a_{k-1}}{(x+a_1) \cdots (x+a_k)}$

Hey guys I was reading Alfred van der Poortens paper regarding Apery's constant and I came across this pretty equality. For all $a_1, a_2, \ldots$ \begin{align} \large\sum_{k=1}^{\infty}\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots (x+a_k)} = \frac…
Achal
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How do I derive the formula for the sum of squares?

I was going over the problem of finding the number of squares in a chessboard, and got the idea that it might be the sum of squares from $1$ to $n$. Then I searched on the internet on how to calculate the sum of squares easily and found the below…
anirudh
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a divergent series which $\prod 1+a_n$ converges.

i am searching for a series with this condition that $\prod 1+a_n$ converges but $\Sigma a_n$ diverges. i know that if $a_n = n^{\frac{1}{2}}$ then $\Sigma a_n$ diverges but i dont know it is exactly what i want, does $\prod 1+a_n$…
user115608
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Dedekind and Bois-Reymond criterion

1)Dedekind criterion If $\lim a_n=0$, $\sum_{n=1}^{\infty} (a_n-a_{n-1})$ converges absolutely and the partial sums of $\sum_{n=1}^{\infty} z_n$ are bounded, then $\sum_{n=1}^{\infty} a_nz_n$ converges. 2)Bois-Reymond criterion If…
user100106
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Prove uniform convergence and identity for the series $\sum^{\infty}_{n=0} e^{-nx}$ and $\sum^{\infty}_{n=0} ne^{-nx}$ where $x > 0$.

Prove uniform convergence and identity for the series $\sum^{\infty}_{n=0} e^{-nx}$ and $\sum^{\infty}_{n=0} ne^{-nx}$ where $x > 0$. Consider $\sum^{\infty}_{n=0} e^{-nx}$ and $\sum^{\infty}_{n=0} ne^{-nx}$ where $x > 0$. I've proved that…
Shuzheng
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How to prove uniform convergence for sequences $f_n = x^n(1-x), f_n = \frac {n^3x} {1+n^4x^2}$ and $ f_n = \sqrt {n} xe^{-nx^2}$ on $[0,1]$

How to prove uniform convergence for sequences $f_n = x^n(1-x), f_n = \frac {n^3x} {1+n^4x^2}$ and $ f_n = \sqrt {n} xe^{-nx^2}$ on $[0,1]$ I've already shown that the following sequence of functions converge pointwise to $0$ on $[0,1]$: $$f_n =…
Shuzheng
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Any hint on this summation problem?

Given that $μ$ and $Q$ are real constants and $i$ is a positive integer, evaluate $$\sum_{i=1}^{+\infty}\;i\,\tan^{-1}\left(\frac{\mu Q}{\mu^2+\left(i^2-\frac{1}{4}\right)Q^2}\right)$$
Bhaskar Dey
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Help with Maclaurin Series

I am working on finding a Maclaurin series for this function. $$f(x) =x^6e^{x^7}$$ So I think I have to evaulate the above function based on a Maclaurin series for $e^x$ = $\sum_{n=0}^\infty {x^n\over n!}$ I am just confused on how to connect the…
user140749
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Prove $\sum\limits_{n\mathop=0}^{\infty}\frac{1}{n^2+1}=\frac{1}{2}+\frac{\pi}{2}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$

How do I show that $$\sum\limits_{n\mathop=0}^{\infty}\frac{1}{n^2+1}=\frac{1}{2}+\frac{\pi}{2}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$
user85798
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Calculating the sequence sum

Calculate the sum of n elements of the sequence $a_n$, in which: $a_1=3$, $a_2=33$, $a_3=333$, $a_4=3333$ and so forth. We see that it's not an arithmetic progression as 3333-333 is not equal to 333-33 and so on. It also isn't geometric…
Bringiton
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Is the "reverse" sum summable?

Suppose we have a summable series $\sum_{n=0}^{\infty}a_n$, where $a_n$ are positive. Define new $b_n=\sum_{k=0}^{\infty} a_k-\sum_{k=0}^na_k$. Does $\sum_{n=0}^{\infty} b_n$ exist?
mpiktas
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Simplifying a series

I have a series like this: $$1 + \frac 1 {x^2} + \frac 1 {x^4} + \frac 1 {x^6} ....$$ Is this a known series? Can I simplify this to something? Thanks.
bdhar
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Sum of a simple infinite series

Evaluate: $$\sum\limits_{n=1}^\infty \frac{n^2}{3^n}.$$ By the ratio test, $\displaystyle\lim_{n\to\infty} \frac{(n+1)^2}{3^{n+1}}\cdot\frac{3^n}{n^2}=1/3,$ which is less than 1, therefore the series is convergent. Now I am stuck on how to…
codeedoc
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