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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Finding the common ratio of a geometric series from the sum and first term

If the sum of a geometric series is 80, and the first term is 5, and the number of terms is 5, how can I determine the common ratio?
Orkwad
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Series of a function

Consider $$f(x)=\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$$ For what values of $x$ does the series converge absolutely? For values $x>0$ we…
user43758
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Let $z \in \mathbb C$, $|z| = 1$. Assume the sequence $a_n = z^n$ is convergent. Prove $z = 1$.

Let $z \in \mathbb C$, $|z| = 1$. Assume the sequence $a_n = z^n$ is convergent. Prove $z = 1$. The case $z = 1$ implies convergence of $a_n$ is easy to prove. It is also easy to prove that $z = -1$ implies divergence of $a_n$. So clearly $z$ must…
Shuzheng
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Is there an expression using the main constants of mathematics as result of the following infinite sum:

$$\sum_{k=0}^\infty {{\pi^{k\over 2}}\over {\Gamma({k\over 2} +1)}}$$ I've found, that $\sum_{k=0}^\infty {{\pi^{k\over 2}}\over {\Gamma({k\over 2} +1)}}$ = $e^{\pi} + 2\sum_{k=0}^\infty {{({4\pi})^{k}k!}\over {(2k+1)!}}$ But does that help?
Redundant Aunt
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Limit of $\frac{(\sum_{1} ^ n a_j)^p}{n^{p - 1} \sum_{j = 1} ^ n a_j ^ p}$ where $\frac{\sum a_j}{n} \to \infty$

Let $a_1, a_2, ...$ be a sequence of positive numbers such that $\frac{\sum_{j = 1} ^ n a_j}{n} \to \infty$ as $n \to \infty$. What can we say about the behavior of $$\frac{(\sum_{j = 1} ^ n a_j)^p}{n^{p - 1} \sum_{j = 1} ^ n a_j ^ p}$$ as $n \to…
guy
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Radius and Interval of Convergence of $\sum (-1)^n \frac{ (x+2)^n }{n2^n}$

I have found the radius of convergence $R=2$ and the interval of convergence $I =[-2,2)$ for the following infinite series: $\sum_{n=1}^\infty (-1)^n \frac{ (x+2)^n }{n2^n}$ Approach: let $a_n = (-1)^n \frac{ (x+2)^n }{n2^n}$ Take the ratio…
theta
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What is the sum of (n-1)+(n-2)+...+(n-k)?

What is the sum of this series ? $(n-1)+(n-2)+(n-3)+...+(n-k)$ $(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$ So how can we find the sum from $n-1$ to $n-k$ ?
ammar
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Solutions to equation involving trigonometry and geometric series

$$\cos^2 x + \cos ^3 x +\dots = 1+ \cos x$$ I want to find values of $x$ between $0$ and $ 180$ degrees for which the above equation holds true. Attempt at a solution: left side is a converging geometric progression, for which $a_1$ is $\cos^2 x$…
Bak1139
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evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n}$

I am trying to compute the sum $$\sum_{n=1}^\infty \frac{n^2}{3^n}.$$ I would prefer a nice method without differentiation, but if differentiation makes it easier, then that's fine. Can anyone help me? Thanks.
dami
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When is the formula for the infinite geometric series valid

When is the formula $$S_{\infty} = \dfrac{a}{1-r}$$ valid? Does |$r| <1$?
Phaptitude
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How do I prove that the series $\sum_{n=1}^\infty \frac{1}{n}$ diverges?

I am studying infinite series in class. Our teacher showed us how the series $$\sum_{n=1}^\infty \frac{1}{n}=1+1/2+1/3+...$$ diverges because $$\int_{1}^\infty \frac {1}{n}dn=\infty$$ Is there a better way to show how the above series diverges?
GTX OC
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More series help, this time a telescopic sum.

$$\sum_{n=2}^\infty \frac{2}{n^2 - 1}.$$ I Tried setting it up as a telescoping sum as $$\frac{2}{n} - \frac{2}{n-1}.$$ but now i'm sure that cannot be correct. Mayhaps I need to complete the square? or pull out an N? I'm sorry for asking so many…
Tibbiticus
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What is the series $\sum_{i=n}^{\infty}\frac{i}{2^i}$

When computing the expected value for a random variable I reached the following series: $$\sum_{i=n}^{\infty}\frac{i}{2^i}$$ I am confident it is convergent, but have no idea how to compute it.
V G
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Function of the following sequence

I feel like this should be obvious for me, but I am not understanding this one. Given: 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,..., I need to find a function for this. Thank you.
DezTop
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How prove a sequence has infinitely many square numbers.

Question Suppose $u$ and $v$ are positive integers and $\{a_n\}$ is a sequence with $a_{1}=u+v$ and $$\begin{cases} a_{2m}=a_{m}+u\\ a_{2m+1}=a_{m}+v, \end{cases}$$ for every $m \ge 1$. Let $s_{m}=a_{1}+a_{2}+\cdots+a_{m}$, show that…
math110
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