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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Convergence of $\sum\limits_{n=1}^\infty \sin^2(\pi(n + \frac{1}{n})) $?

My friend was practicing for his entrance examination and one of the problems on older exams was this one. It asks if it converges and to explain the reasoning behind that answer. The series is: $$\sum_{n=1}^\infty \sin^2\left(\pi(n +…
hadsed
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What's the result of $\sum_{k=1}^{\infty} {k \cdot \left[ (1-2^{-k})^n - (1-2^{-k+1})^n \right]}$?

Came from the problem "Expectation of maximum times of doing $n$ times of flipping coin tests until getting the reverse side". The probability of maximum times of flipping among $n$ independent tests should be: $$ \begin{align} \text{Pr}(X = 1) &=…
Daniel Wang
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How find the closed -form $\{a_{n}\}$ if such Strange condtion

let $\{a_{n}\}$ such $a_{1}=0,a_{n}<0,\forall n\ge 2$,such $$(a_{n+1}+1)^2(2a_{n+1}+1)=2a^4_{n+1}(2a_{n}+1)(a_{n}+1)$$ find the closed form $a_{n}$ it easy to get $$a_{2}=1-\sqrt{2},~~~~a_{3}=3+2\sqrt{2}-\sqrt{2(10+7\sqrt{2})}$$ also by condtion we…
math110
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Show that the sequence $a_n=1-\frac13+\frac{1}{3^2}-...+(-1)^n\frac{1}{3^n}$ is bounded.

Show that the sequence ${a_n}$ where $$a_n=1-\dfrac13+\dfrac{1}{3^2}-...+(-1)^n\dfrac{1}{3^n}$$ is bounded. The first thing that came to my mind was to see if the sequence is monotone. If I am right, the $(n+1)th$ term should look like this:…
kormoran
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Partial sums of the logarithmic function

Consider the following $$\int^a_0 \frac{1-x^k}{1-x }\, dx \text{ on the disk } |a|\leq 1.$$ This can simplified to $$\sum_{n\geq 0}\int^a_0 x^n (1-x^k) \, dx $$ $$\sum_{n\geq 1}\frac{a^n}{n}-\frac{a^{n+k}}{n+k} $$ $$\sum_{n\geq…
Zaid Alyafeai
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Knock-out tournament: pruning number of teams after each round

Suppose I start with $N$ teams ($N$ is even) which are randomly paired with each other in the first round. $\frac N2$ teams progress into the next round. I want this process to repeat itself until one team remains. But this is problematic if we have…
rjt90
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Show the sequence $(1 - \frac{1}{n})^{-n}$ is decreasing.

How do you show the sequence $(1 - \frac{1}{n})^{-n}$ is decreasing? I understand that the binomial theorem should be used here but I don't see how we can use it to prove that $a_{n+1} < a_n$. I will rewrite the sequence as, \begin{align*} (1 -…
foo
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$a_{m^2}=a_m^2,a_{m^2+k^2}=a_ma_k$ sequence

Sequence $\{a_n\},n\in\mathbb N_+$ with all terms positive integers satisfy $a_{m^2}=a_m^2,a_{m^2+k^2}=a_ma_k$. Find $\{a_n\}$. I suppose all terms of $\{a_n\}$ are $1$. This problem makes me think of a lot of conclusions,…
user1034536
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Convergence of series $\sum\limits_{n=1}^{\infty} \frac{((n+1)!)^n}{2!\cdot 4!\cdot \ldots \cdot (2n)}$

I am trying to show that the following series is absolutely convergent: $$\sum\limits_{n=1}^{\infty} \frac{((n+1)!)^n}{2!\cdot 4!\cdot \ldots \cdot (2n)}$$ After writing the denominator as $\prod\limits_{k=1}^{n}(2k)!$ I have tried applying the…
user1024526
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$\lim (a_n - a_{n-1}) = 0$ for convergent $a_n$

If $a_n$ is convergent and sequence $b_n = a_n - a_{n-1}$, then $\lim b_n = 0$ It's true because $\lim b_n = \lim (a_n - a_{n-1}) = \lim a_n - \lim a_{n-1} = 0$. The last limits are equal due to convergence of $a_n$. Is it correct proof?
mange
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Harmonic series principal value

Harmonic series $$ 1+1/2+1/3+1/4+\cdots $$ is divergent. However if we take the generalized Hurwitz harmonic series $$ F(s)=\sum_{n=0}^{\infty}(n+a)^{-1+s}+\sum_{n=0}^{\infty}(n+a)^{-1-s}$$ can we say or regularize the result so $ F(s) \to -2\Psi…
Jose Garcia
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Shifted alternating harmonic series

Suppose I have the following series $$ \sum_{k=1}^{\infty}(-1)^{k+1}\frac{1}{k+ia}, \hspace{1cm}$$ with $i$ the imaginary unit and $a \in \mathbb{R}$. Does this shifted sum converge?
korni1990
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Two quite messy sequences

Consider the following $50-$term sums$:$ $$ S=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+....\frac{1}{99\cdot100}$$ and $$T=\frac{1}{51\cdot100}+\frac{1}{52\cdot99}+\frac{1}{53\cdot98}+....+\frac{1}{100\cdot51}$$ Express $\frac{S}{T}$ as…
Zootopia
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Grandi's series contradiction

This is the Grandi's series: $1-1+1-1+1-1+\dots$ The series can be equal to $0$ $$(1-1)+(1-1)+(1-1)+\dots=0+0+0+\dots=0,$$ or to $1$ $$1-(1-1)-(1-1)-(1-1)-\dots=1-0-0-0\dots=1,$$ or to $1/2$ $$S=1-1+1-1+1-\dots,\quad\quad …
TurtleLoop
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How does this infinite sum series switch between ln(2) and -(1-ln(2))?

I found this simple infinite series of summed terms: 0/1 + 1/2 - 2/3 + 3/4 - 4/5 + 5/6 - 6/7 ....etc When I tried calculating a bunch of terms, weirdly enough, it seems that an even number of terms produces ln(2) and an odd number of terms produces…
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