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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Reordering sequences

I am trying to study the reordering of sequences. I am having difficulties finding any learned articles, probably because I don't know the key words to search for. I hope if I can describe what I want to do someone can can expand my knowledge or…
OldCurmudgeon
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How to calculate sum for $k\ge 1\in\mathbb N\quad\sum\limits_{i=1}^\infty\frac1{i(i+1)(i+2)...(i+k)}$

$$\forall k\ge 1\in\mathbb N\\\displaystyle\sum\limits_{i=1}^\infty\frac1{i(i+1)(i+2)...(i+k)}=?$$ Try$(1)$I tried to apply vieta If it be considered like this…
Micheal Brain Hurts
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What does this series converge to, if anything?

$$\arctan{1} + \arctan{\frac{1}{2}} + \arctan{\frac{1}{3}} + \arctan{\frac{1}{4}} ...= ?$$ The infinite series for arctan is $$\arctan{x} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} ...$$ So I want to sum up the $\arctan{1\over n}$ where $n$…
DrZ214
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Evaluate $1+\left(\frac{1+\frac12}{2}\right)^2+\left(\frac{1+\frac12+\frac13}{3}\right)^2+\left(\frac{1+\frac12+\frac13+\frac14}{4}\right)^2+...$

Evaluate: $$S_n=1+\left(\frac{1+\frac12}{2}\right)^2+\left(\frac{1+\frac12+\frac13}{3}\right)^2+\left(\frac{1+\frac12+\frac13+\frac14}{4}\right)^2+...$$ a_n are the individual terms to be summed. My Try…
Almot1960
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Is there a closed form formula for $\sum\limits_{n=0}^\infty e^{-n^2}$?

I am curious about the value of $$\sum_{n=0}^\infty e^{-n^2}.$$ It comes to my mind by observing Gauss integral that it is equal to $$\int_{0}^\infty e^{-t^2}dt=\dfrac{\sqrt{\pi}}{2}.$$
Zir
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Show that $\sum_{n=0}^\infty \frac{\sin {nx}}{a^n} = \frac{a \sin{x}}{1 + a^{2} - 2a \cos{x}}$

Show that $$\sum_{n=0}^\infty \frac{\sin {nx}}{a^n} = \frac{a \sin{x}}{1 + a^{2} - 2a \cos{x}}$$ I've been trying to use the geometric series rule for $\sum_{n=0}^\infty x^{-n} = \frac{x}{x -1}$ as well as Euler for the $\sin(nx)$, but I just…
Jason
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Is there any double series that cannot exchange the sum?

I want to find a sequence $(u_{n,p})_{(n,p)\in\mathbb{N}^2}$ that satisfied: $$ \sum_{n=0}^{\infty}\sum_{p=0}^{\infty}u_{n,p} ~\text{is convergent} $$ $$ \sum_{p=0}^{\infty}\sum_{n=0}^{\infty}u_{n,p} ~\text{is convergent…
TimeCoder
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floor number sum

For $a$ and $n$, there is formula to calculate $$a + 2a + 3a + \cdots + na = \frac{n(n+1)}{2} a.$$ Is there formula: $$\lfloor a\rfloor + \lfloor 2a\rfloor + \lfloor 3a\rfloor + \cdots + \lfloor na\rfloor $$
RAM
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A nice infinite series: ${\sum\limits_{n=1}^\infty}\frac{1}{n!(n^4+n^2+1)}=\frac{e}{2}-1$ - looking for a more general method

while clearing out some stuff I found an old proof I wrote for an recreational style problem a while back, here's the sum: ${\sum\limits_{n=1}^\infty}\frac{1}{n!(n^4+n^2+1)}=\frac{e}{2}-1$ I'll show how I got to the limit in a moment, but because my…
Mehness
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Identify $ \sum\limits_{k=-\infty}^{\infty} \frac{1}{(x+k)^2}$

Identify $$ f(x)=\sum\limits_{k=-\infty}^{\infty} \frac{1}{(x+k)^2}.$$ Is this a known expansion of a simpler looking function? I understand that this series becomes infinite at integers.
Srini
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Calculate the sum of the first N terms of the sequence

$a_n=a_{n-1}\displaystyle \frac{n+1}{n}$ if $n > 1$ $a_n=1$ if $n=1$ I'm not too sure where to start here. This is part of a review for a class and I can't really seem to remember what we're reviewing. The first 5 values are... $a_1 =…
Hoser
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Sum of the series $1\cdot3\cdot2^2+2\cdot4\cdot3^2+3\cdot5\cdot4^2+\cdots$

Find the sum of $n$ terms of following series: $$1\cdot3\cdot2^2+2\cdot4\cdot3^2+3\cdot5\cdot4^2+\cdots$$ I was trying to use $S_n=\sum T_n$, but while writing $T_n$ I get a term having $n^4$ and I don't know $\sum n^4$. Is there any other way…
MathGeek
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How come $\lim_{n\rightarrow \infty} \frac{a_n}{a_{n-1}}$ be different than 1?

There is a theorem that "$\forall_{n}: a_n>0 ~and~ \lim_{n\rightarrow \infty} \frac{a_n}{a_{n-1}}=L \Rightarrow \lim_{n\rightarrow \infty} \sqrt[\leftroot{-2}\uproot{2}n]{a_n}=L$. Does the left hand side of the statement also implies that $a_n$ does…
Bush
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Comparison of two convergence conditions for sequences of non-negative numbers

Let $a_n\geq 0$ be a sequence of non-negative numbers. Consider the following two statements: $$ \text{(I)}\qquad\qquad \lim_{n\to\infty} \frac{1}{n^2}\sum_{i=1}^n a_i =0, $$ $$ \text{(II)}\qquad\qquad\qquad \sum_{n=1}^\infty…
Rasmus
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Write the general term of the periodic sequence $1$, $-1$, $-1$, $1$, $-1$, $-1$, $1$, ..., as $(-1)^{g(n)}$ or other closed form

How to put mathematically sequence that changes sign like: $n = 0\quad f = 1$ $n = 1 \quad f = -1$ $n = 2 \quad f = -1$ $n = 3 \quad f = 1$ $n = 4 \quad f = -1$ $n = 5 \quad f = -1$ $n = 6 \quad f = 1$ ..... In the form of (-1)^(something) or…
Anonymous
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