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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T/F: If $\sum_{n=1}^{\infty}a_{n}$ converges, then $\sum_{n=1}^{\infty}(a_{n})^{3}$ converges.

My math GSI proposed this question to the class today, and even he doesn't know the answer! (This should be answered separately for absolute and conditional convergence)
rosstex
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How to prove the series $\sum_{n=1}^{\infty} \frac{\sin(\frac{1}n)}n$ converges?

$$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}n)}n$$ Using the comparison test/limit comparison test? I have tried the comparison test and several attempts at the limit comparison test, but everything I try points to divergence, which I know isn't true.
rosstex
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Infinite series for partial sums of square roots.

Can you prove these infinite series for partial sums of square roots? $$\sqrt{1}=\sum\limits_{n=1}^{\infty }…
Mats Granvik
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How prove this series is converges? $\sum_{n=1}^{\infty}\dfrac{1}{n\ln{(n^3+n)}}$

this series $$\sum_{n=1}^{\infty}\dfrac{1}{n\ln{(n^3+n)}}$$ is converge ？ I conside $$\dfrac{x}{1+x}<\ln{(1+x)}
math110
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Sum of a series of minimums

I should get sum of the following minimums.Is there any way to solve it? $$\min\left\{2,\frac{n}2\right\} + \min\left\{3,\frac{n}2\right\} + \min\left\{4,\frac{n}2\right\} + \cdots + \min\left\{n+1, \frac{n}2\right\}=\sum_{i=1}^n \min(i+1,n/2)$$
user16948
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Find the $n^{th}$ term for this sequence

I have the following sequence: 4, 7, 11, 19, 36, 69 Now, I've done the usual and found the differences, and it goes down to four levels until I get a common difference, suggesting I have to use $n^{4}$ somewhere, but I just can't find the nth…
Some Guy
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Certain arithmetic progessions in the Cantor set

How to prove or disprove that the Cantor set does not include any arithmetic progression of length 5?
Markiyan Hirnyk
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Show that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if $|r| < 1$

Possible Duplicate: Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$ Show, rigorously, that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if |r| < 1. Also, show that if |r| < 1, the sum is given…
UGPhysics
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closed form sum: $\sum_{i=1}^n \frac{2^{-i}}{i}$

I came across this gnarly sum and was wondering if anyone knew any tricks to get a closed form for it: $$\sum_{i=1}^n \frac{2^{-i}}{i}$$
user47098
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Value of $\sum\limits_{n= 0}^\infty \frac{n²}{n!}$

How to compute the value of $\sum\limits_{n= 0}^\infty \frac{n^2}{n!}$ ? I started with the ratio test which told me that it converges but I don't know to what value it converges. I realized I only know how to calculate the limit of a…
Maik Klein
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How Find this $T_{20}-S_{20}$

Let $\{a_{n}\},\{b_{n}\}$ be sequences such that $$\sqrt{\dfrac{a_{n}+1}{b_{n}}}=\dfrac{1}{n}, \quad S_{n}=\sqrt{T_{n-1}} \quad(n>1)$$ where $\displaystyle S_{n}=\sum_{i=1}^{n}a_{i}$, $\displaystyle T_{n}=\sum_{i=1}^{n}b_{i}$, and $b_{1}=1$.…
math110
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How to show the series $\displaystyle\sum_{\xi\in\mathbb Z^n}\frac{1}{(1+|\xi|^2)^{s/2}}$ converges if and only if $s>n$?

Let $s\in\mathbb R$. How to show the series $$\sum_{\xi\in\mathbb Z^n}\frac{1}{(1+|\xi|^2)^{s/2}}$$ converges if and only if $s>n$ ($n$ is the dimension of $\mathbb Z^n$)? The convergence of this series is to be understood as the existence of the…
PtF
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Sums of logarithms $\small \log(1+1/1) + \log(1+1/2) + \ldots \log(1+1/n) $ how is this telescoping?

This is surely a tiny question but I seem to have some blackout currently ... I tried to define a function for the sum of logarithms, like we have it for the sums of like powers with the bernoulli-polynomials. (I had a question with sums of…
Gottfried Helms
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Sum of a power series with a parameter

Does anybody know the sum $$ \sum_{n=0}^{\infty}\frac{x^{n}}{(n+a)n!}=f(x,a)$$ here $a$ is a number $ a>0 $. A hint please ? :D If $ a=1$ I believe $ f(x,1)=(e^{x}-1)/x $
Jose Garcia
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Proof of $\frac{1}{(1-r)^2}$

I was given that $S=1+2r+3r^2+\cdots = \frac{1}{(1-r)^2}$ was an identity from my students, and I tried to prove directly using geometric series. I got stuck and looked around online only to be told that taking the derivative of the series that…
hyg17
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