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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infinite series involving harmonic numbers and zeta

I ran across a fun looking series and am wondering how to tackle it. $$\sum_{n=1}^{\infty}\frac{H_{n}}{n^{3}}=\frac{{\pi}^{4}}{72}.$$ One idea I had was to use the digamma and the fact that…
Cody
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Find sum of series $\sum_{n=1}^{\infty}\frac{1}{n(4n^2-1)}$

I need help with finding sum of this: $$ \sum_{n=1}^{\infty}\frac{1}{n(4n^2-1)} $$ First, I tried to telescope it in some way, but it seems to be dead end. The only other idea I have is that this might have something to do with logarithm, but really…
SekstusEmpiryk
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Find the summation $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \cdots$

What is the value of the following sum? $$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \cdots$$ The possible answers are: A. $e$ B. $\frac{e}{2}$ C. $\frac{3e}{2}$ D. $1 + \frac{e}{2}$ I tried to expand the options using the series…
Taylor Ted
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Every sixth polynomial shares a factor of $(a^2-6)$

I currently looking at the polynomials you get from the series expansion of $$ \frac{1-x^2}{1-ax+2x^2}=1+a x +(a^2-3) x^2 +(a^3-5a) x^3 +\underbrace{(a^4-7a^2+6)}_{(a-1)(a+1)(a^2-6)} x^4+\dots $$ W|A helped here... What I found is that from…
draks ...
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"Proof" that $1-1+1-1+\cdots=\frac{1}{2}$ and related conclusion that $\zeta(2)=\frac{\pi^2}{6}.$

Sorry if this has been posted before. Can somebody please tell me whether this result is correct, and give explanation as to why or why not? I'm not good at the formal side of maths. Start here:…
Hobbyist
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Consider convergence of series: $\sum_{n=1}^{\infty}\sin\left[\pi\left(2+\sqrt{3}\right)^n\right]$

Consider convergence of series: $$\sum_{n=1}^{\infty}\sin\left[\pi\left(2+\sqrt{3}\right)^n\right]$$ My tried: We have $$\sum_{n=1}^{\infty }\sin(\pi…
Iloveyou
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If $\sum a_n$ converges, prove $\sum a_n^3$ converges

I can prove this if $a_n \geq 0$ for all $n$ (by comparison test). How should one tackle this in general? Moreover, is it true that the converse also holds?
Covi
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Is there an alternating series that satisfies only one of the conditions of the Alternating Series Test that nonetheless converges?

I was recently helping a college math student with her homework. Her teacher had offered an extra-credit question: Find two alternating series $\sum_{n=1}^\infty (-1)^{n-1}a_n$ such that $a_{n+1} \leq a_n$ for all $n$, but $\lim_{n\to\infty} a_n…
Sean
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Is there a sequence such that $\sum{a_n}$ diverges but $\sum{na_n}$converges?

Is there a (real) sequence such that $\sum{a_n}$ diverges but $\sum{na_n}$converges?
Maddy
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Showing that $\sum\limits_{n \text{ odd}}\frac{1}{n\sinh\pi n}=\frac{\ln 2}{8}$

$$-\frac{8\varepsilon_0V_0}{\pi}\sum_{n \text{ odd}}\frac{1}{n\sinh(n\pi)}=\boxed{\displaystyle-\frac{\varepsilon_0V_0}{\pi}\ln 2.}$$ I have not found a way to sum this series analytically. Mathematica gives the numerical value $0.0866434$, which…
erik
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Does this sequence always give a square number?

Question : Supposing that a sequence $\{a_n\}$ is defined as $${a_{n+3}}^2=-{a_{n+2}}^2+2{a_{n+1}}^2+48a_{n+1}a_{n}+32{a_n}^2\ (n\ge 1)$$ $$a_1=a_2=a_3=1$$ then, is $a_n$ a square number for any $n$? For example, we can see $$\sqrt{a_n} :…
mathlove
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Number of occurrences of the digit 1 in the numbers from 0 to n

We have a function: $f(n)$ = number of occurrences of the digit $1$ in the numbers from $0$ to $n$. Example: $f(12) = 5$ It is obvious that $f(1)=1.$ Question: Which is the next number for which $f(n) = n$?
Mark Zimmers
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Limit of $\frac 12$, $\frac{1\cdot 4}{2 \cdot 3}$, $\frac{1\cdot 4\cdot 5}{2\cdot 3\cdot 6}$, ...

The following is not a homework, just curiosity. Consider the integers grouped by consecutive pairs : $(1,2)$, $(3,4)$, ... What is the limit of the "switching fractions" where we alternatively use the largest number in a pair either upward or…
Thomas Sauvaget
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Ramanujan-Sums... How do they do?

Prove that: $\displaystyle \frac{1}{e^{2\pi}-1}+\frac{2}{e^{4\pi}-1}+\frac{3}{e^{6\pi}-1}+...=\frac{1}{24}-\frac{1}{8\pi}$ I would like to see different ways of solving this beautiful sum, whoever is encouraged? :) Thanks to all.
Dek
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show this $\lceil \frac{n}{1-a_{n}}\rceil =n+1$

let $a_{n}$ be squence such $a_{1}=2-\dfrac{\pi}{2}$, and $$a_{n+1}=2a_{n}+\dfrac{a_{n}-1}{n},n\in N^{+}$$ show that $$f(n)=\lceil \dfrac{n}{1-a_{n}}\rceil =n+1$$ I try:since $$f(1)=\lceil \dfrac{1}{\dfrac{\pi}{2}-1}\rceil=2$$ and…
math110
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