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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A limit with Product and Cosine

Can anyone help me please to find the limit of this series: $$\lim_{x \to 0}\frac{\displaystyle\prod_{k=1}^{n}\left(1-\sqrt[k]{\cos x}\right)}{x^{2n-2}}$$ Thanks for all :D
Sherloek holmes
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Show that the following series is convergent.

Consider the series whose general term is as follows: $$u_n=\frac{a_n}{(S_n)^\lambda}$$ with the condition $S_n = \sum_{k=1}^{n}a_k$ with constraints that $0\leq a_n\leq 1,$ $S_n$ is a divergent series and $\lambda >1.$ Show that the series is…
Student
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How do I know the limit of this infinite sequence

I have $a_k=\frac1{(k+1)^\alpha}$ and $c_k=\frac1{(k+1)^\lambda}$, where $0<\alpha<1$ and $0<\lambda<1$, and we have a infinite sequence $x_k$ with the following evolution equation. $$ x_{k+1}=\left(1-a_{k+1}\right)x_{k}+a_{k+1}c_{k+1}^{2} $$ I…
user8827
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Does $\sum\limits_{k=1}^\infty \sum\limits_{n=k}^\infty \frac{(-1)^{n+k}}{n}$ diverge?

Does $\displaystyle\sum_{k=1}^\infty \sum_{n=k}^\infty \frac{(-1)^{n+k}}{n}$ diverge? It is clear that the alternating Harmonic series converges: $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}=\log 2.$$ Thus, $S_k=\displaystyle\sum_{n=k}^\infty…
user519208
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Then value of $\alpha^2 +4\alpha$ in Infinite series

If $\displaystyle \alpha = \frac{5}{2!\cdot 3}+\frac{5\cdot 7}{3!\cdot 3^2}+\frac{5\cdot 7 \cdot 9}{4!\cdot 3^3}+\cdots \cdots \infty.$ Then value of $\alpha^2 +4\alpha$ is Try: Let $$S = \frac{5}{2!\cdot 3}+\frac{5\cdot 7}{3!\cdot…
DXT
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How to find closed form for a partial infinite product?

I ran across this infinite product: $$\lim_{n\to\infty}\prod_{k=2}^n\left(1-\frac1{\binom{k+1}{2}}\right)$$ I easily found that it converges to 1/3. Using my calculator, I found that $$1-\frac1{\binom{k+1}{2}}=\frac{(k-1)(k+2)}{k(k+1)}$$ Then, here…
Cody
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What is the closed form for $ \sum\limits_{n=0}^{\infty}\frac{1}{(n!)^2}$?

What is the closed form for $\sum_{n=0}^{\infty}\frac{1}{(n!)^2}$? And is there a closed form for $\displaystyle \sum_{n=0}^{\infty}\frac{1}{(n!)^k}$? Edit: If there are no closed forms for the two series, how should I convert them into integrals?…
Larry
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Evaluating $\frac{1^2}{2^1}+\frac{3^2}{2^2}+\frac{5^2}{2^3}+\cdots$ using sigma notation

This question can be solved by method of difference but I want to solve solve it using sigma notation: $$\frac{1^2}{2^1}+\frac{3^2}{2^2}+\frac{5^2}{2^3}+\cdots+\frac{(2r +1)^2}{2^r}+\cdots$$ I used the geometric progression summation for the…
jame samajoe
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How do you show that $\sum_{n=0}^\infty\frac1{n!(n+2)}=1$?

I've been trying to understand the result of this sum: $$\sum_{n=0}^\infty\frac1{n!(n+2)}=\frac12+\frac13+\frac18+\frac1{30}+\frac1{144}+\dots=1$$ Could you show me how to obtain 1 as result?
André Rasera
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How I find limit of $P_n$

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n=2,3,\dots$. Define $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\dots\left(1+\frac{1}{a_n}\right)$$Find $$\lim_{n \to \infty}P_n$$ Trial:…
A.D
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Convergence of the series $\sum\limits_{n=1}^{\infty }(-1)^{n+1} \frac{\sin^2(n)}{n}$

I would like to know how to prove the convergence (or not) of the following serie: $\sum\limits_{n=1}^{\infty }(-1)^{n+1} \frac{\sin^2(n)}{n}$ Thank you in advance for any suggestion.
Ilda Reis
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I don't know whether the sequence converges

We have $a_0 = x_2\in [0,A]$ and $a_{n+1} = (A−a_n)/2$. Prove that this sequence converges to $A/3$. And then prove the same for the sequences $a_{2k}$ and $a_{2k+1}$.
Elena
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How to solve $\frac{1}{1000.1998}+\frac{1}{1001.1997}+\cdots+\frac{1}{1998.1000}$

The question is: If $$A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\cdots+\frac{1}{1997.1998}$$ and $$B=\frac{1}{1000.1998}+\frac{1}{1001.1997}+\cdots+\frac{1}{1998.1000}$$ then what is the value of $\frac{A}{B}$? I could figure out that…
Fasal123
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Evaluate $\sum\limits_{n=1}^{\infty} \frac{x^{2^n}}{1-x^{2^n}}$ when $x<1$

Where does $\displaystyle\sum_{n=1}^{\infty} \frac{x^{2^n}}{1-x^{2^n}}$ converges to (for some $x \in \mathbb{R}$ with x<1)? Since we failed with the Ansatz here, we were able to proof that the series diverges for $x>1$ and converges for $x<1$ by…
mjb
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How fast does $\sum_{k=1}^n \frac{1}{k^\alpha}$ ( $\alpha\leq1$) diverge?

The series $\sum_{k=1}^\infty \frac{1}{k^\alpha}$ diverges if $\alpha\leq 1$. How can I estimate the divergent rate when $\alpha$ is given. For example, if $\alpha=1$, $\sum_{k=1}^n \frac{1}{k^\alpha}=O(\log n)$; if $\alpha=0$, $\sum_{k=1}^n…
Wayson Kong
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