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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Limit of $\prod_{i=1}^n (1-1/2^i)$

I am trying to find the limit of the sequence $$s_n:=\displaystyle \prod_{i=1}^n \left(1-\frac{1}{2^i} \right)$$ The sequence is decreasing and bounded below by $0$. I guess that the limit is $0$, is there any way to show this ? Or, is there any…
pritam
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Is there any partial sums of harmonic series that is integer?

is there any partial sums of harmonic series that add up to an integer? partial sums not as trivial as the first term only i.e. 1, or the powers of 2 i.e the infinite geometric series for 2. This might be trivially obvious for a subset of…
jimjim
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Infinite sum of $1/n^n$

I'm interested in the sum of what I'll call inverse self powers over integers, namely $$\sum_{n=1}^{\infty}\frac{1}{n^n}$$ Almost by accident I found that $$\sum_{n=1}^{\infty}\frac{1}{n^n}=\int_0^1\frac{\mathrm{d}x}{x^x}$$ which is a pretty neat…
Vinsanity
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Closed form of $\sum_{n=0}^{\infty }\frac{1}{2^n(4n-1)}$

Closed form of $$\sum_{n=0}^{\infty }\frac{1}{2^n(4n-1)}$$ the Wolfram gave me the answer below, but I ask if there is a clear value of this sum
E.H.E
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To check convergence of series $\sum_{1}^{\infty}\frac{\ln(n)}{n(n+1)}$

I have to check whether series is convergent or not. $$\sum_{n=1}^{\infty}\frac{\ln(n)}{n(n+1)}$$ I used condensation test for this so i get new series as $$ \frac{n \ln(2)}{2^n(2^n+1)}.$$ Now I apply ratio test for new series and I get $\lim_{n…
Taylor Ted
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Let $f(x) = (x^n-1)/(x-1)$. Why does $f(1)=n$?

In the back of De Souza (Berkeley Problems in Mathematics, page 305), it says: For $x \neq 1$, $$ f(x) = (x^n-1)/(x-1) = x^{n-1} + \cdots + 1 $$ so $f(1) = n$. The expansion for $x \neq 1$ obviously follows from the definition of a partial…
mathjacks
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Find $\lfloor 1000S \rfloor$ for $S =\sum_{n=1}^{\infty} \frac{1}{2^{n^2}} = \frac{1}{2^1}+\frac{1}{2^4}+\frac{1}{2^9}+\cdots.$

Let $$S = \displaystyle \sum_{n=1}^{\infty} \dfrac{1}{2^{n^2}} = \dfrac{1}{2^1}+\dfrac{1}{2^4}+\dfrac{1}{2^9}+\cdots.$$ Find $\lfloor 1000S \rfloor$ My attempt was to recognize that consecutive perfect squares increase in the form…
user19405892
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In a general definition, a sequence starts at zero or at one?

In calculus textbooks we read that a sequence is a function whose domain is the set of positive integers. While in french textbooks we read that a sequence is a function whose domain is the set of non-negative integers. My question is about…
palio
  • 11,064
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Convergence of primes over factorials

What is the value of $\displaystyle \sum_{n=1}^{\infty} \dfrac{p_i}{i!} =\dfrac{2}{1!}+\dfrac{3}{2!}+\dfrac{5}{3!}+\dfrac{7}{4!}+\cdots$? This question really interested me since we all know that $\displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n!} =…
Jacob Willis
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Calculating the value of $\sum^\infty_{k=1}\frac{1}{k(9k^2-1)}$

My first thought is to split it up into: $$\sum^\infty_{k=1}\;\frac{3}{2(3k-1)} + \frac{3}{2(3k+1)} - \frac{1}{k}$$ This is starting to look vaguely like some sort of rearrangement of the alternating harmonic series with some factors thrown in. The…
pizzaroll
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Is there a mathematical proof that shows all multiples of $5$ either end with a $0$ or $5$?

I know that all multiples of $5$ end up with a $0$ or $5$ as the last digit. But there are an infinite amount of numbers. Is there a way to formally prove that this is true for all numbers using variables?
Mark
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Simplification of sums

Sorry for this uninteresting question but hopefully someone can provide some help. Is there a way to simplify the following expression? $$\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\frac{n-m-v}{n}\right)^{r}\displaystyle…
r_31415
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Computing $\sum_{n=0}^{\infty} u_{n}, \frac{u_{n+1}}{u_n}=\frac{n+a}{n+b}$

I would like to evaluate $$ \sum_{n=0}^{\infty} u_{n}$$ where $u_{n}$ is defined by the following recurrence relation: $$ \frac{u_{n+1}}{u_n}=\frac{n+a}{n+b}$$ $$ a,b>0$$ As $$ \frac{u_{n+1}}{u_n}=1-\frac{b-a}{n}+o(1/n) $$ a sufficient condition for…
Chon
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Ratio of two Sequences converging to zero

If $\{x_{n}\}$ is a sequence of positive real numbers, $0
Paul
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Three Sequences

Given the sequences $a_{n},b_{n},c_{n}$ all subsets of the interval $(0,1)$, and all converges to 0. Also we have $a_{n}\leq b_{n}^{\alpha}$, and $a_{n}\leq c_{n}$, for all $n=1,2,3,...$, for some $\alpha \geq 1$ (real number). Now, for a given…
Alpha
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