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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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An exercise about convergence of series

Possible Duplicate: Convergence/divergence of $\sum\frac{a_n}{1+na_n}$ when $\sum a_n$ diverges. Let $a_n$ be a non-negative sequence such that the series $\sum a_n$ does not converge. Could the series $\sum a_n/(1+na_n)$ be convergent?
user25640
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Solving without induction show that $a_{n}=2n-1$

Let $a_{1}=1$,and such $$4S_{n}=n(a_{n}+a_{n+1})$$ where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ find $a_{n}$ since $a_{2}=3$,and we can easy to prove $a_{n}=2n-1$ Induction Methods Assume $a_{k}=2k-1$, so we…
partofsha
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convergence of alternating series — weakening a hypothesis

A comment below this answer inspires this question. Suppose $a_n\in\mathbb{R}$ for $n=1,2,3,\ldots$ and $|a_n|\to0$ as $n\to\infty$. Further suppose the terms alternate in sign. If moreover the sequence $\{|a_n|\}_{n=1}^\infty$ is decreasing, then…
Michael Hardy
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If $A=\frac{1}{\frac{1}{1980}+\frac{1}{1981}+\frac{1}{1982}+........+\frac{1}{2012}}\;,$ Then $\lfloor A \rfloor\;\;,$

If $$A=\frac{1}{\frac{1}{1980}+\frac{1}{1981}+\frac{1}{1982}+........+\frac{1}{2012}}\;,$$ Then $\lfloor A \rfloor\;\;,$ Where $\lfloor x \rfloor $ Represent floor fiunction of $x$ My Try:: Using $\bf{A.M\geq H.M\;,}$ We…
juantheron
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Solving linear recursive equation $a_n = a_{n-1} + 2 a_{n-2} + 2^n$.

I wish to solve the linear recursive equation: $a_n = a_{n-1} + 2a_{n-2} + 2^n$, where $a_0 = 2$, $a_1 = 1$. I have tried using the Ansatz method and the generating function method in the following way: Ansatz method First, for the homogenous…
user279515
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Finding the sum to n terms of series :$\frac{1}{1\cdot 2\cdot 3\cdot 4} +\frac{1}{2\cdot 3\cdot 4\cdot 5} + \frac{1}{3\cdot 4\cdot 5\cdot 6}+\cdots$

$$ \frac{1}{1\cdot 2\cdot 3\cdot 4} +\frac{1}{2\cdot 3\cdot 4\cdot 5} + \frac{1}{3\cdot 4\cdot 5\cdot 6}+\cdots $$ up to $n$ terms. I need help in solving this sum. I tried finding the coefficients of terms after splitting the terms..: it…
Riyaa
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Evaluating $\frac{1}{\sqrt{4}+\sqrt{6}+\sqrt{9}}+\frac{1}{\sqrt{9}+\sqrt{12}+\sqrt{16}}+\frac{1}{\sqrt{16}+\sqrt{20}+\sqrt{25}}=?$

It's a question (not hw) I bumped into few years back. Couldn't make any real progress with. Maybe you can help? $$\frac{1}{\sqrt{4}+\sqrt{6}+\sqrt{9}}+\frac{1}{\sqrt{9}+\sqrt{12}+\sqrt{16}}+\frac{1}{\sqrt{16}+\sqrt{20}+\sqrt{25}}=?$$ Thanks.
Amihai Zivan
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Partial sum of $\sum \frac {1} {k^2}$

It is pretty well-known that $\sum_{k = 1}^{\infty} \frac {1} {k^2} = \frac {\pi^2} {6}$. I am interested in evaluating the partial sum $\sum_{k = 1}^{N} \frac {1} {k^2}$. Here is what I have done so far. Since we have $$\sum_{k = 1}^{N} \frac {1}…
user98186
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Showing $\sum _{k=1} 1/k^2 = \pi^2/6$

Possible Duplicate: Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$ Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$? I read my book of EDP, and there appears the next serie $$\sum _{k=1}…
Jonathan Prieto-Cubides
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How can I prove $\ e^n=\sum_{k=0}^\infty\frac{n^k}{k!}$

Given $$ e=\sum\limits_{k=0}^\infty\frac{1}{k!} $$ How can I prove $$ e^n=\sum\limits_{k=0}^\infty\frac{n^k}{k!} $$ Can anyone please demostrate the $n=2$ case? Thanks!
zjk
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why this sequence is $a_n=2n-1?$

Let postive integers sequence $\{a_n\}$,if for any postive integers $m$ we have $$\sum_{i=1}^{a_m}a_{i}=(2m-1)^2$$ show that $$a_n=2n-1$$ It seem can use mathematical induction? Now only following $$\sum_{i=1}^{a_m}…
user253631
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A "naive" proof approach for the convergence of $(n!)^2/(2n)!$

Proof that $(n!)^2/(2n)!$ converges to $0$. I take following steps: $(n!)^2/(2n)(2n-1)\cdots(n!) = (n!)/(2n)(2n-1)\cdots(n-1)$. I assume (do I need to prove?) that $n!$ divides $(2n)(2n-1)\cdots(n-1)$. So I have at the end $1/K$ ($K$ is the…
Ignace
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"Complement" of Kempner Series

It is a long time since I summed any series. I was aware that the harmonic series diverged, (if I recall you can keep making groups that are greater than a half). Then today I saw SMBC and it blew my mind. http://smbc-comics.com/index.php?id=3777 A…
Luskentyrian
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How do you prove $\sum \frac {n}{2^n} = 2$?

How do you prove $$\sum_{n=1}^{\infty} \frac {n}{2^n} = 2\ ?$$ My attempt: I have been trying to find geometric series that converge to 2 which can bind the given series on either side. But I am unable to find these. Is there a general technique to…
saubhik
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Proof- central term of recursive pyramid

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. How can we prove $4^{-n}(n+1)\left(2^{2n+1}-{2n+1\choose n+1}\right)$ produces the central term in the $2n+1$th row…
Ali
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