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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Sum $\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n(n^4+4)}$

I want to evaluate the sum $$\large\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n(n^4+4)}.$$ I did partial fraction decomposition to get $$\frac{1}{2^n}\left(\frac{-1}{n^2+2n+2}+\frac{4}{n^2-2n+2}+1\right)$$ I am absolutely stuck after this.
sn24
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Sum of a series of a number raised to incrementing powers

How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+\cdots+2^n$. What math course deals with this sort of calculation? Thanks much!
Julian A.
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value of $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\cdots$?

Is there anything known about the value of the series $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\frac{1}{1+2+3+4+5+6}+\cdots$ ?
jimjim
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Odd-Odd-Even-Even Sequence

I want a sequence that alternates between being an even integer and being an odd integer and I've come up with this sequence $ s_n=\lfloor \frac{n}{2} \rfloor $. So, it goes $0,1,1,2,2,3,3,\ldots$ and I was wondering if I can do something similar…
russjohnson09
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How to prove $ \sum_{r=1}^{k-1} \binom{k}{r}\cdot r^r \cdot (k-r)^{k-r-1} = k^k-k^{k-1} $

How to prove the following identity: $$ \sum_{r=1}^{k-1} \binom{k}{r}\cdot r^r \cdot (k-r)^{k-r-1} = k^k-k^{k-1} $$ I have no idea how to tackle it because of the $r^r$. Any help is highly appreciated!
Redundant Aunt
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Sum of infinite series and polynomial equation

Let $$a=\sum_{n=1}^\infty(1 \mod \phi^{-n})2^{n/3},$$ where $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio and $1 \mod x=1-x\lfloor x^{-1}\rfloor$. How can I prove that $a$ satisfies the equation $$2255 a^6 - 2340 a^5 - 3174 a^4 - 672 a^3 +…
Vladimir Reshetnikov
  • 47,122
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Find the Sum of the Series: $1/(x+1) + 2/(x^2 + 1) + 4/(x^4 +1) +\cdots$ $n$ terms

Find the sum of $n$ terms the following series: $$\frac1{x+1} + \frac2{x^2 + 1} + \frac4{x^4 +1} +\cdots\qquad n\text{ terms}$$ $t_n$ seems to be $\dfrac{2^{n-1}}{x^{2^{n-1}} + 1}$ But after that I'm not sure as to how to proceed
Rohan
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Calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$

How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?
wonderingdev
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Nested recurrence sequence with interesting properties

This is my first post here, thanks for stopping by. The question as written below comes from the book 'A Concise Introduction To Pure Mathematics'. I've included my working (this isn't a homework problem) and points I am interested in -…
WierdFishes
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Find a sum of appropriate values of $\cos$ and $\sin$ to determine the value of a series

The task is to find a sum of multiple values $\cos$ and $\sin$ to determine the value of $$\sum_{n=1}^\infty (-1)^n \frac{n}{(2n+2)!}$$ Since I had no clue how to approach this I consulted Wolfram|Alpha which returned this…
Lenar Hoyt
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Does this series diverge: $(\sqrt 2-\sqrt 1)+(\sqrt 3-\sqrt 2)+(\sqrt 4-\sqrt 3)+(\sqrt 5-\sqrt 4)+\dots$?

$$(\sqrt 2-\sqrt 1)+(\sqrt 3-\sqrt 2)+(\sqrt 4-\sqrt 3)+(\sqrt 5-\sqrt 4)…$$ I have found partial sums equal to natural numbers. The first 3 addends sum to 1. The first 8 sum to 2. The first 15 sum to 3. When the minuend in an addend is the square…
Jeffrey Young
  • 841
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Convergent or divergent $\sum\limits_{n=0}^{\infty }{\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n+2)}}$

\begin{align} & \sum\limits_{n=0}^{\infty }{\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n+2)}} \\ & \text{ordering} \\ & a_{n}=\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n+2)}=\frac{1\cdot 3\cdot 5...(2n-3)(2n-1)\cdot…
lazlo
  • 123
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Does the series converge or diverge?

I want to check, whether $$\sum\limits_{n=0}^{\infty }{\frac{n!}{(a+1)(a+2)...(a+n)}}$$ converges or diverges. $a$ is a constant number Ratio test $$\begin{align} & \frac{a_{n}}{a_{n-1}}=\frac{n!}{(a+1)(a+2)...(a+(n-1))(a+n)}\cdot…
lazlo
  • 123
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represent $\cos(x^{1/2})$ by Maclaurin series

I need to represent $\cos(x^{1/2})$ by Maclaurin series. I'm not sure that what I have done is correct. We know Maclaurin series for $$\cos(x) = 1- {x^2 \over 2!} +{x^4\over 4!}+\cdots$$ So I substitue $x^{1/2}$, and I got $$\cos(x^{1/2})= 1 -…
Adam Sh
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How find $a_{n}$ if the sequence $a_{n}=2a_{n-1}+(2n-1)^2a_{n-2},n\ge 1$

Question: let sequence $\{a_{n}\}$,such $a_{-1}=1,a_{0}=1$ and $$a_{n}=2a_{n-1}+(2n-1)^2a_{n-2},n\ge 1$$ Find the $a_{n}$ I find $$a_{0}=1,a_{1}=3,a_{2}=15,a_{3}=105$$ and I found $$a_{0}=1$$ $$a_{1}=1\cdot 3$$ $$a_{2}=1\cdot 3\cdot…
math110
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