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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Convergence of Infinite Series

How can I determine that the following series is convergent: $$ \sum_{x = 1}^{\infty}\left(\,\sqrt[3]{\,{x^{3} + 1}\,}\, - x\right) $$ I used the limit divergence test and I found that the limit of the nth term is zero. So that was of no use.…
Eddy
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$\sum_{n=1}^\infty n\beta_n<\infty$ implies $\sum_{n=1}^\infty n\alpha_n\beta_n<\infty$, for $\alpha_n\to\infty$?

Suppose that $\beta_n$ is a sequence of positive real numbers satisfying $$\sum_{n=1}^\infty n\beta_n<\infty$$ Is it possible to find a sequence $\alpha_n$ of positive numbers such that $\alpha_n\to\infty$ and $$\sum_{n=1}^\infty…
Tomás
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what is 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 +1/9 - ...?

I know that it is converging because it is alternating series with terms getting smaller to zero. but I do not know what it converges to value
rhh
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Closed Form for an Infinite Summation

Let $p$ be prime, and let $0\le j\le p-1$. I would like to find a closed expression for the infinite sum $$f(p,j)=\sum_{i=0}^{\infty}\frac{p^i}{(pi+j)!}$$ Initial computations show that $f(2,0)=\cosh\sqrt2-1$ and $f(2,1)=\frac{\sinh\sqrt 2}{\sqrt…
Jared
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Convergence of a Cauchy Product

I have the following series obtained via the Cauchy Product of the alternating harmonic series with itself $\left( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \right ) \cdot \left( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \right ) = \sum_{n=1}^{\infty}…
VinalV
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Is it possible to represent the natural number "1" as the sum of p-series in this way?

My argument: $$1=(\frac{1}{2})^2+(\frac{1}{3})^2+\cdots+(\frac{1}{2})^3+(\frac{1}{3})^3+\cdots=\sum_{k=2}^\infty (\frac{1}{k})^2+\sum_{k=2}^\infty (\frac{1}{k})^3+\cdots .$$ Explanation) First, for any natural number $n\geq2$, the following holds:…
mathvoca
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showing that $\sum_1^n \frac{1}{\sqrt{n^2+k}}$ is increasing

Define $u_n$ as : $$u_n=\displaystyle\sum_{k=1}^n \dfrac{1}{\sqrt{n^2+k}}$$ From first values, $(u_n)$ seems to be increasing. From squeezing, its limit is $1$. $$ \dfrac{1}{\sqrt{n^2+n}} \le \dfrac{1}{\sqrt{n^2+k}} \le \dfrac{1}{\sqrt{n^2+1}}$$…
ahmed
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The closed form of $\sum_{k=1}^{\infty}\left(\frac{2}{3}\right)^{k^2}\frac{3^k}{2^k-3^k}$

Are you kind to show me the way? I want to find its closed form. $$\sum_{k=1}^{\infty}\left(\frac{2}{3}\right)^{k^2}\frac{3^k}{2^k-3^k}$$
Don'tdothesmart
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$\sum_{n=1}^\infty \frac{1}{3^n-2^n}$

I'm trying to calculate the following series: $$ \sum_{n=1}^\infty \frac{1}{3^n-2^n}=\frac{1}{3-2}+\frac{1}{9-4}+\ldots $$ or the general case: $$ \sum_{n=1}^\infty \frac{1}{(x+1)^n-x^n} $$ I have tried some methods but still have no idea, is there…
3usi9
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Prove convergence of two series:

I would like to prove if the next two series are convergent. First: $$ \sum_{n=1}^{\infty}\log\left(\frac{n+1}{n}\right)\arcsin \left(\frac{1}{\sqrt{n}}\right) $$ I think that this series is convergent, so $$\arcsin\left(\frac{1}{\sqrt{n}}\right)$$…
user711823
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Prove that there are integers $n$ such that $\sqrt 2[f(1) + \cdots + f(n)] \in \mathbb Z$, $f(x) = \sqrt{1 - \frac{1}{2x^2 + \sqrt{4x^4 + 1}}}$.

Define function $f(x)$ as followed $$\large f(x) = \sqrt{1 - \dfrac{1}{2x^2 + \sqrt{4x^4 + 1}}}, \forall x \in \mathbb Z^+$$. Prove that there are infinitely many positive integers $n$ such that $$\large \sqrt 2[f(1) + f(2) + \cdots + f(n - 1) +…
Lê Thành Đạt
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Sum of series with addition: $\sum_{n=1}^\infty \frac{1}{n^2(n+1)}$

I am looking at some homework where I have: $\sum_{n=1}^\infty \frac{1}{n^2(n+1)}$ How can I sum this? I know that $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$ and also that $\sum_{n=1}^\infty \frac{1}{n(n+1)}=1$ But I just can't see or…
Daniel
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Of the first 100 terms in fibonacci sequence, how many are odd?

So, as I tried to solve this questions I used the following : $\newcommand{\aa}{\mathbf{a}}$ $\aa_1 =1,~~ \aa_2 = 1$ and we know that in Fibonacci sequence, $\aa_n = \aa_{n-1} +\aa_{n-2}. $ Hence, $\aa_3 = 2,~~ \aa_4 = 3,~~ \aa_5 = 5,~~ \aa_6 =…
Jimjamlorde
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Value of $\sum\limits^{\infty}_{n=1}\frac{\ln n}{n^{1/2}\cdot 2^n}$

Here is a series: $$\displaystyle \sum^{\infty}_{n=1}\dfrac{\ln n}{n^{\frac12}\cdot 2^n}$$ It is convergent by d'Alembert's law. Can we find the sum of this series ?
Laura
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How to get the following summation of the series $\sum\limits_{n=0}^{\infty}\frac{1}{n!(n^4+n^2+1)}$

I am trying to find the sum $$\sum\limits_{n=0}^{\infty}\frac{1}{n!(n^4+n^2+1)}$$ I had factorized the sum as $$\frac{1}{2n(n!)}\left(\frac{1}{n(n-1)+1}-\frac{1}{n(n+1)+1}\right)$$ From this step, how to proceed?
vqw7Ad
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