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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How to calculate $\sum\limits_{n=1}^{\infty}\frac{1}{3^n - 2^n}$

By the test of reason, this series converges. The problem is figuring out which technique to use to calculate your sum. Thanks for any help.
Mathsource
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Infinite series problem

The sum of $$\frac{2}{4-1}+\frac{2^2}{4^2-1}+\frac{2^4}{4^4-1}+\cdots \cdots $$ Try: write it as $$S = \sum^{\infty}_{r=0}\frac{2^{2^{r}}}{2^{2^{r+1}}-1}=\sum^{\infty}_{r=0}\frac{2^{2^r}-1+1}{(2^{2^r}-1)(2^{2^r}+1)}$$ d not know how to solve from…
DXT
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How to find $\sum_{n=0}^{+\infty} \dfrac{1}{(kn)!}$?

I want to calculate $$S_k = \sum_{n=0}^{+\infty} \dfrac{1}{(kn)!}$$ when $k\in\Bbb N ^*$. I tried to find a recurrence equation for $k$, but I found nothing really interesting. I already know that for $k=1, S_1 = e$ and $S_2 = ch(1)$, but I don't…
MiKiDe
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Find the largest $n \in \mathbb{N_+} $ such that $\{ (2+\sqrt 2)^n\} < \frac{7}{8}$, where $\{x\}$ denotes the fractional part of $x$.

Problem Find the largest $n \in \mathbb{N_+} $ such that $\{ \left(2+\sqrt{2}\right)^n\} < \dfrac{7}{8}$, where $\{x\}$ denotes the fractional part of $x.$ My Solution First, we can prove that $a_n=(2+\sqrt{2})^n+(2-\sqrt{2})^n$ is an integer…
mengdie1982
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Example of a series!

Give an example of a convergent series $$\sum_{n=1}^{\infty}a_n$$ such that the series $$\sum_{n=1}^{\infty}a_{3n}$$ is divergent.. I cannot find find any kind of series... I am also be very thankful if you find a divergent series (changing the…
Auro Das
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Kind of counter intuitive sum of log gamma

I came across an infinite series that appears to be rather counter intuitive. Show that $\displaystyle \sum_{k=1}^{\infty}(-1)^{k+1}\ln(\Gamma(k+1))=\frac{-1}{4}\ln\left (\frac{\pi}{2}\right)$ At first glance, it obviously diverges. I ran this…
Cody
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Prove $\lim_{m\to\infty}\sum_{k=1}^m\frac{2^{-k}}{k} = \log 2$

While working on a harder double sum, I (erroneously) reduced it to the sum below, which I recognized numerically to rapidly converge to $\log 2$. Prove $$\lim_{m\to\infty}\sum_{k=1}^m\frac{2^{-k}}{k} = \log 2$$ The cute observation is that if you…
Mark Fischler
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sum of logarithms

I have to solve find the value of $$\sum_{k=1}^{n/2} k\log k$$ as a part of question. How should I proceed on this ?
user8250
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If $\sum\limits_{i=1}^na_i=\prod\limits_{i=1}^na_i$ for every $n$, identify $\lim\limits_{n\to \infty}a_n$

Let $\left(a_n\right)_{n \in\mathbb{N}} $ denote a sequence of real numbers such that, for every $n\geqslant1$, $$\sum_{i=1}^na_i=\prod_{i=1}^na_i$$ Identify the limit $$\lim_{n\to \infty}a_n$$ What I have done: $$a_1-a_1=0 \\a_1+a_2-(a_1a_2)=0…
Fricul38
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The integer part of a sequence

I was trying to solve the following sum: $$a_1=\sqrt[3]{24}$$ $$a_{n+1}=\sqrt[3]{(a_n+24)},n\ge1$$ $Find\,the\,integer\,part\,of\,a_{100}$ Source: ISI B Math 2012 paper I proceeded in this…
Saurav Goyal
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Find the smallest possible value of $a_1$.

Let $a_1,a_2,...,a_{11}\in \mathbb{N}$ with $a_1
JSCB
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Find the sum of the series $S = \sum_{k=1}^{n} \frac{k}{k^{4} + k^{2} + 1} $

$$ S = \sum_{k=1}^{n} \frac{k}{k^{4} + k^{2} + 1} $$ I started by factorizing the denominator as $k^2+k+1$ and $k^2-k+1$ The numerator leaves a quadratic with $k$ and $k-1$ or a constant with $k+1$ and $k-1.$ I tried writing the individual terms,…
CodeBlooded
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How many Arithmetic Progressions having $3$ terms can be made from integers $1$ to $n$?

How many Arithmetic Progressions having $3$ terms can be made from integers $1$ to $n$? The numbers in the AP must be distinct. For example if $n=6$ then number of AP's possible are $6$ $1,2,3$ $2,3,4$ $3,4,5$ $4,5,6$ $2,4,6$
PRYM
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If a series $\sum\lambda_n$ of positive terms is convergent, does the sequence $n\lambda_n$ converge to $0$?

Let $\lambda_n>0, n\in\mathbb{N}$, with $\sum_n \lambda_n<+\infty$. Can I conclude that $n\lambda_n\to 0$? In this question and this question and their answers, it is shown that this is true if $\lambda_n$ are decreasing. What happens if $\lambda_n$…
Anton
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How to calculate $\sum_{r=1}^\infty\frac{8r}{4r^4+1}$?

Calculate the following sum: $$\frac{8(1)}{4(1)^4+1} + \frac{8(2)}{4(2)^4+1} +\cdots+ \frac{8(r)}{4(r)^4+1} +\cdots+ \text{up to infinity}$$ MY TRY:- I took $4$ common from the denominator. and used $a^2+b^2=(a+b)^2-2ab$. It gave me two brackets,…
Priyanka Vithani
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