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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Why does this sum $\sum\limits_{n=1}^{\infty}{\frac{2^{2^{n-1}}}{2^{2^n}-1}}$ converge to 1?

A friend of mine gave me this quiz: Where does this sum converge to? $$\sum\limits_{n=1}^{\infty}{\frac{2^{2^{n-1}}}{2^{2^n}-1}}$$ $a_1=\frac{2}{3} , a_2=\frac{4}{15}, a_3=\frac{16}{255},...$and $S_1=\frac{2}{3}, S_2=\frac{14}{15} ,…
John. P
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$1+ {1\over 11}+ {1\over 111}+ {1\over 1111}+....=?$

What is the sum of the series $$1+ {1\over 11}+ {1\over 111}+ {1\over 1111}+....$$.The partial sum is a monotonically increasing and bounded above sequence, so sum must exits in real.
Supriyo Halder
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Evaluate $\sum\limits_{n=0}^{\infty}(-1)^n\sum\limits_{j=0}^{k}{k \choose j}\frac{(-1)^j}{2n+2j+1}$

Evaluate $g(k)$ if $\sum\limits_{n=0}^{\infty}(-1)^n\sum\limits_{j=0}^{k}{k \choose j}\frac{(-1)^j}{2n+2j+1}=\frac{\pi}{2^{2-k}}+g(k)$. The well-known Gregory Series, $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}=\frac{\pi}{4}\tag1$$ Let us generalize…
Endgame
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Evaluating $\sum^{\infty}_{k=1}\frac{k^2}{(2k-1)(2k)(2k+1)(2k+2)}$

Finding sum of series $$\displaystyle \sum^{\infty}_{k=1}\frac{k^2}{(2k-1)(2k)(2k+1)(2k+2)}$$ Try: Let $$S = \displaystyle \sum^{\infty}_{k=1}\frac{k^2}{(2k-1)(2k)(2k+1)(2k+2)}$$ So, $$S =\sum^{\infty}_{k=1}\frac{k^2\cdot…
DXT
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show this sum is $\sum_{n=0}^{+\infty}|a_{n}|<+\infty$

define sequence $a_{n}$ if $a_{0},a_{1}$ be arbitrary real number ,and such $$a_{n}=a_{n-1}-\dfrac{2}{n}a_{n-2}$$ show that $$\sum_{n=0}^{+\infty}|a_{n}|<+\infty$$ I remember seeing someone asking this question before.But I can't find it,can you…
math110
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Series Help: $\sum_{n>0}\frac{1}{n^{1+|\sin n|}}$

Possible Duplicate: Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge? I am having a very hard time with the following series: $$\sum_{n>0}\frac{1}{n^{1+|\sin n|}}$$ I don't even have clues for…
Heitor Fontana
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Is my solution to this sequence problem correct?

Here is a Olympiad Problem and i have a solution for it already , please tell me know if i will get full marks for my solution or not (i think my solution is short than official solution)? You can also post your alternative solutions ^_^ Let…
Maggi Iggam
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How can I determine general formula of this sequence?

I am trying to find general formula of the sequence $(x_n)$ defined by $$x_1=1, \quad x_{n+1}=\dfrac{7x_n + 5}{x_n + 3}, \quad \forall n>1.$$ I tried put $y_n = x_n + 3$, then $y_1=4$ and $$\quad y_{n+1}=\dfrac{7(y_n-3) + 5}{y_n }=7 -…
minhthien_2016
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Does assumimg $1+2+4+\cdots$ is summable in any sense make any contradiction?

Some of diverging series is Cezaro- or Abel-summable. There are some good properties of Cezaro and Abel summation: (1) inserting finitely many zeroes does not change the value, (2) interchanging finitely many terms does not change the value, (3) it…
J1U
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How to prove $a_{n} < 2$ if $\displaystyle a_{n+2}=a_{n+1}+\frac{a_{n}}{3^n}$

Let $(a_{n})_{n \geq1}$ be a real sequence such that $a_{1}=a_{2}=1$ and $\displaystyle a_{n+2}=a_{n+1}+\frac{a_{n}}{3^n}, n\geq 1$. Prove that $a_{n} < 2, \forall n \geq 1.$ I write $$\sum a_{k+2}-a_{k+1}=\sum \frac{a_{k}}{3}$$ and I obtained :…
Iuli
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Sequence of arithmetic means of a square summable sequence is square summable

Is it true that if $x \in \ell^2$ then $\left(\frac{1}{n} \sum_{i=1}^n x_i\right)_{n} \in \ell^2$ ? I conjecture that this is false and the sequence $x_n = \frac{1}{\sqrt{n}\ln(n)}$ is a counterexample, but I cannot prove it.
Andrei Kh
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$\sum_{n=0}^{\infty}\frac{x^{n+3}}{(n+3)!}$

I am stuck summing a simple infinity series $$\sum_{n=0}^{\infty}\frac{x^{n+3}}{(n+3)!}$$ I know that $$\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=e^x$$ And I presume I should divide my expresion into some kind partial fractions right? Something like…
Sarunas
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last 2 digits of a sequence

$x+\frac{1}{x} = 3$, what are the last 2 digits of $x^{2^{2013}}+\frac{1}{x^{2^{2013}}}$? Getting the next value, we have to square then subtract by 2, I am clueless in getting to the next step
SuperMage1
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Arithmetic series problem

Given $\left\{a_n\right\}$ arithmetic progression, $a_1=2$, $a_{n+1}=a_n+2n$ $\left(n\:\ge \:1\right)$. $a_{50}=?$ What i did: $$a_n+d=a_n+2n$$ $$d=2n$$ $$a_{50}=2+d\left(n-1\right)$$ $$a_{50}=2+2\left(n^2-n\right)$$ $$a_{50}=2+2\cdot…
Serif Yaohim
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