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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What is the closed-form for $\displaystyle\sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + mn+41n^2}$?

Omitting the case $m = n = 0$, if, $$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{2\pi \ln\big(\tfrac{5 + \sqrt {29}}{\sqrt2}\big)}{\sqrt {58}} $$ as in this post, then is, $$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2…
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Showing a series is convergent.

Possible Duplicate: Contest problem about convergent series Let ${p}_{n}\in \mathbb{R} $ be positive for every $n$ and $\sum_{n=1}^{∞}\cfrac{1}{{p}_{n}}$ converges, How do I show that…
Leitingok
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Evaluating $\int\frac{x^b}{1+x^a}~dx$ for $a,b\in\Bbb N$

In this previous answer, MV showed that for $n\in\Bbb N$, $$\int\frac1{1+x^n}~dx=C-\frac1n\sum_{k=1}^n\left(\frac12 x_{kr}\log(x^2-2x_{kr}x+1)-x_{ki}\arctan\left(\frac{x-x_{kr}}{x_{ki}}\right)\right)$$ where $$x_{kr}=\cos…
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Is there a closed form for $\sum_{n=0}^{\infty}{2^{n+1}\over {2n \choose n}}\cdot\left({2n-1\over 2n+1}\right)^2?$

We have $$\sum_{n=0}^{\infty}{2^{n+1}\over {2n \choose n}}\cdot{2n-1\over 2n+1}=4-\pi\tag1$$ I would like to know if there exist a closed form for $$\sum_{n=0}^{\infty}{2^{n+1}\over {2n \choose n}}\cdot\left({2n-1\over 2n+1}\right)^2 =\,??\tag2$$ I…
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What is the general term formula of this recurrence relation, if it exists?

$$a_n=-\frac{(a_{n-1} + 1)^2}{a_{n-1}+2}\quad \quad a_1 = -\frac 12$$ I transformed to: $$a_n = -a_{n-1} - \frac{1}{a_{n-1} + 2}$$ $$a_{n-1} = -a_{n-2} - \frac{1}{a_{n-2} + 2}$$ ... And then $$a_n - a_{n-1} + a_{n-2} -\cdots + \cdots -…
Boyang
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The sequence $a_1 = \frac{1}{2}, a_2 = 1, a_{n+1} = \frac{na_n+1}{a_{n-1}+n}$ is decreasing

Consider the sequence $\{a_n\}$ defined by $$a_1 = \frac{1}{2}, a_2 = 1, a_{n+1} = \frac{na_n+1}{a_{n-1}+n}, \forall n\ge 2.$$ Prove that $\{a_n\}_{n\ge 3}$ is decreasing. I get the first $200$ values of $\{a_n\}$ and recognize this fact, but I…
Tien Kha Pham
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Explicit value for $\sum_{n=1}^{\infty} \left(\frac{1}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}\right)$

This question came out from this other one: Is there an explicit value for this series? $$\sum_{n=1}^{\infty}\frac{1}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}= \sum_{n=1}^{\infty}…
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Does the alternating sum of prime reciprocals converge?

$$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime. I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits:…
Robert Soupe
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Solve summation $\sum_{i=1}^n \lfloor e\cdot i \rfloor $

How to solve $$\sum_{i=1}^n \lfloor e\cdot i \rfloor $$ For a given $n$. For example, if $n=3$, then the answer is $15$, and it's doable by hand. But for larger $n$ (Such as $10^{1000}$) it gets complicated . Is there a way to calculate this…
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Regarding the sum $\sum_{p \ \text{prime}} \sin p$

I'm very confident that $$\sum_{p \ \text{prime}} \sin p $$ diverges. Of course, it suffices to show that there are arbitrarily large primes which are not in the set $\bigcup_{n \geq 1} (\pi n - \epsilon, \pi n + \epsilon)$ for sufficiently small…
MathematicsStudent1122
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Prime one heap Nim

I have been working on an interesting problem my lecturer mentioned recently. Prime Nim is a variant of the Nim game where you have a single pile with an arbitrary number $n\in \Bbb N+\{0\}$ of elements and players can take away a prime count of…
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$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$ by integration or differentiation

There is an infinite sum given: $$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$$ It should be solved using integration, derivation or both. I think using power series can help but I don't know how to finish the calculation. Any help will be appreciated!
Hendrra
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Evaluation of $\sum _{n=1}^{\infty} \tan^{-1} \frac{2}{n^2+n+4}$

Find the following sum $$S= \sum _{n=1}^{\infty} \tan^{-1} \frac{2}{n^2+n+4}$$ I am not able to make it telescopic series. Could someone help me with this?
Mathematics
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Prove that if $x_{n+2}=\frac{2+x_{n+1}}{2+x_n},$ then $x_n$ converges

Let $x_0 > 0$, $x_1 > 0$ and $$x_{n+2} = \dfrac{2+x_{n+1}}{2+x_n}$$ for $n \in \{0,1,\dots\}$. Prove that $x_n$ converges. My attempt: If $x_n$ converges and $\lim\limits_{n\rightarrow\infty}x_n=a$ so $a=\frac{2+a}{2+a}$, which gives $a=1$. Now,…
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Is the following limit finite ....?

I would like to see some clue for the following problem: Let $a_1=1$ and $a_n=1+\frac{1}{a_1}+\cdots+\frac{1}{a_{n-1}}$, $n>1$. Find $$ \lim_{n\to\infty}\left(a_n-\sqrt{2n}\right). $$
DIEGO R.
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