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.

65378 questions
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${a_n}$ for a sequence containing no zeroes

Take the sequence of Natural numbers which do not contain the digit zero. So your sequence becomes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12 ... Can we find an expression for ${a_n}$ ?
user71408
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Sums like $\sum_{n=0}^{\infty}\frac{1}{a^{n}+b^{n}}$?

Is there a trick for calculating sums like $$ S(a,b) := \sum_{n=0}^{\infty}\frac{1}{a^{n}+b^{n}} $$ where $a$ and $b$ are constants? I've run through my usual bag of tricks: reducing it to a series I already know, telescoping, realizing the sum as a…
Lopsy
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Find the sum of the n first series numbers: $7,77, 777,...$

Find the sum of the $n$ first numbers: $7,77, 777,...$ I thought to find an order by dividing $77/7=11, 777/7=111...$ but I don't know how to continue.
seda
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Proving convergence of sum of two convergent series'.

Suppose $$\sum_{n=1}^\infty a_n~,~\sum_{n=1}^\infty b_n$$ are convergent series' how could I use this to prove that $$\sum_{n=1}^\infty \frac{a_n+b_n}{2}$$ is also convergent using the definition for convergence of a series? Thanks
Ryan
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Evaluating $\sum\limits_{n=1}^{\infty} \left(\frac{1}{4 n-3}+\frac{1}{4 n-1}-\frac{1}{2 n}\right)$

$$\sum\limits_{n=1}^{\infty} \left(\frac{1}{4 n-3}+\frac{1}{4 n-1}-\frac{1}{2 n}\right) = \;?$$ I have been trying to see if it can be written as sum of two telescope terms but it looks tricky. Any help ?
lilly
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Proving $\sum_{n=0}^{\infty }\frac{\sin^4(4n+2)}{(2n+1)^2}=\frac{5\pi ^2}{16}-\frac{3\pi }{4}$

I dont have an idea to prove it because of exist $\sin(4n+2)^4$ $$\sum_{n=0}^{\infty }\frac{\sin^4(4n+2)}{(2n+1)^2}=\frac{5\pi ^2}{16}-\frac{3\pi }{4}$$
user189855
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If $\sum_{n=1}^{\infty} a_n$ converges, then so does $\sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$

Given a positive sequence $(a_n)$. Prove that if $\displaystyle \sum_{n=1}^{\infty} a_n$ converges, then so does $\displaystyle \sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$. I tried to use the root test, and I was stuck with the case $\lim \sup…
Tien Kha Pham
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How is $S_{n-1} = n^2 +2$?

I'm learning Arithmetic Progression. There's an example given in my book which I'm not able to understand from yesterday. The example is: If the sum to $n$ terms of a sequence is given by $S_n =n^2+2n+3$, find $t_n$ and hence find $t_1$ and…
Yogesh Tripathi
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Adding two convergent series

If $\sum_{n=1}^{\infty} a_n$ is finite and $\sum_{n=1}^{\infty} b_n$ is also finite, why is it that you can add the two series term by term and get the sum of the two series? Surely this is reordering the series.
user208567
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sum of even and odd terms in exponential Taylor series

We know for $x \in \mathbb{R}$ that $$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, $$ but what if we were to split the series into the the series containing the even powers of $x$ and the one containing the odd powers of $x$? I.e., $$ \sum_{m=0}^\infty…
bcf
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Is there a closed-form of $ \sum_{n=0}^{\infty }\frac{(-1)^n}{(2n+1)(2n+2)(2n+3)(2n+4)}$

Is there a closed-form of $$\sum_{n=0}^{\infty }\frac{(-1)^n}{(2n+1)(2n+2)(2n+3)(2n+4)}$$ Thanks for any help
user189855
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Evaluating the sum : $\;\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$

How to evaluate this sum? $$\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$$ Please give some technique. Binomial not working.
Learnmore
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A curious expression involving the nearest-integer function whose sum appears to be 3

Let $ \def\nint#1{\langle #1\rangle}\nint x$ denote the integer closest to $\sqrt x$. This is ambiguous whenever $\sqrt x$ is a half-integer; fortunately such will not arise in the rest of this question, and we may simply take $\nint x =…
MJD
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How to prove $\sum_{n=1}^{\infty}\frac{1}{16n^2-16n+3}=\frac{\pi}{8}$

Wolfram alpha computes $$\sum_{n=1}^{\infty}\frac{1}{16n^2-16n+3}=\frac{\pi}{8}$$ But I don't have any idea to prove this. Thank you.
Kong
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If a,b,c are in AP and $a^2,b^2,c^2$ are in HP, then prove either $a=b=c$ or $a,b,- \frac c2 $ are in GP

As the title says. Although first part of the proof is obvious, I'm still able to prove it. And for the second part, I'm essentially trying to prove $b^2=-c/a$ (which is possible only when c<0 Xor a<0). The relations found by me are:…
Harshal Gajjar
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