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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Formula for the sequence repeating twice each power of $2$

I am working on some project that needs to calculate what $a_n$ element of the set of numbers $$1, 1, 2, 2, 4, 4, 8, 8, 16, 16 \ldots$$ will be. $n$ can be quite big number so for performance issues I have to calculate formula for this set of…
Leri
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Is $x_{n+1}=\frac{x_n}{2}-\frac{2}{x_n}$ bounded?

Consider the sequence: $x_1=3, x_{n+1}=\frac{x_n}{2}-\frac{2}{x_n}$. This sequence is bounded or is unbounded? Attempt Checking a few terms we get $x_1 = 3, x_2 = \dfrac{5}{6}, x_3 = -\dfrac{119}{60},x_4 = \dfrac{239}{14280},\cdots$. I will prove…
Puzzled417
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Offset Alternating Series

I have the following alternating series that I would like to determine whether it is absolutely convergent, conditionally convergent, or divergent: $$ \sum\limits^{\infty}_{n=1} \frac{1+2(-1)^n}{n} $$ I have applied some tests and I find it…
Hemofelicity
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Showing $(1-\frac{1}{n}) - (1-\frac{1}{n})^n$ is an increasing sequence?

I've come across a familiar math expression in my research, and I want to formally prove the following. $s_{n+1} - s_{n} > 0$ holds for integer $n \geq 2$, where $s_n := (1-\frac{1}{n}) - (1-\frac{1}{n})^n$. I've plotted this using software, and it…
PurplePenguin
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Evaluate $\sum\limits_{n=0}^\infty\frac{4^n}{(2^{2^{n}}+1)^2}$

I have known that $\displaystyle\sum_{n=0}^\infty\dfrac{2^n}{x^{2^{n}}+1}=\dfrac{1}{x-1}$ for $x>1$. Taking the derivative of both sides , $\displaystyle\sum_{n=0}^\infty\dfrac{4^nx^{2^n-1}}{(x^{2^{n}}+1)^2}=\dfrac{1}{(x-1)^2}$ . This…
Bless
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Can all real numbers be represented by the sum of a convergent series?

I've come up with an example that I think can represent every non-zero rational number: $$\sum_{n=0}^\infty \frac{k}{mx^n}$$ where $k\in \mathbb{Z}_{\neq0}$, $m \in \mathbb{N}_{\neq0}$ and $x \in \Bbb Z \setminus \{ -1, 0, 1 \}$. And I know there…
Marijn
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convergence to infinity of$ (2^n + 3^n)^{\frac{1}{n}}$ (as $n \to \infty$)

Just checking. $2^n$ ($n \to \infty$) tends to $\infty$. + $3^n$ also ($n \to \infty$) tends to $\infty$ so the sum gets me $\infty$. Now $(\infty)^{1/\infty}$ : $(\infty)^0 = 1$ I see no other way. Theorem: $n^{\frac{1}{n}} = 1$ as $n \to…
Ignace
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how do I find the general term here?

I am getting crazy on this series! I found this in a handwritten old book without a reference. I could not figure out how it is built but the series numerically seems to converge to…
Math-fun
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Limit of a sequence

Let $\{u_n\}$ be a sequence of nonnegative numbers satisfying the condition $$ \tag{1} u_{n+1}\leq (1-\alpha_n)u_n+\beta_n \quad \forall n\in\mathbb{N}, $$ where $\{\alpha_n\}$ and $\{\beta_n\}$ are sequences of real numbers such…
drmath
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find Limit $a_n= \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}$

show sequence $a_n= \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}$ converges to log2 my attempt: sequence $a_n$ is monotonic increasing 0<$a_n$<1/2 how to find limit?
ketan
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Does the series $\sum\limits_{n=1}^\infty \frac{1}{n^3 \sin^2n}$ converge or diverge?

I have this series given by $$\sum\limits_{n=1}^\infty \frac{1}{n^3 \sin^2n}.$$ I know that the terms are all nonnegative. Can I get a subsequence of $\frac{1}{n^3 \sin^2n}$ such that its sum diverges to $+\infty$?
OKPALA MMADUABUCHI
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$p$-series divided by alternating $p$-series = geometric series? Why?

I thought the following equation was interesting: $\dfrac{1 + \frac{1}{2^p} + \frac{1}{3^p} + ... }{1 - \frac{1}{2^p} + \frac{1}{3^p} - ...} = \dfrac{1}{1-2^{1-p}}$ for $p>1$, where $p$ is a real number. So in other words, this ratio on the LHS is…
Braindead
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Find the value of $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+ ....... }}}}$

I have tried solving this question. It would be great if someone gives some idea about how to go about solving this question.
Pankaj Gupta
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show that $\sum_{n=1}^{\infty}\frac{1}{n+1}\sum_{k=0}^{n}(-1)^{k+1}\binom{n}{k}\log(k+1)=1$

$$1=\sum_{n=1}^{\infty}\frac{1}{n+1}\sum_{k=0}^{n}(-1)^{k+1}\binom{n}{k}\log(k+1)$$ I tried by several ways but failed -_- the expression is another form to e number by take exponential of both side
mnsh
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Hard recursion involving sums

Let $n$ be an arbitrary integer. Define: $$\begin{align} c_0 &= 1;\\ c_m &= \frac1m \sum_{k = 1}^m (k n - m + k) \frac{(-1)^k}{(2k)!} c_{m - k}. \end{align}$$ This recursion turns up in my quest of computing integrals of functions of Bessel…
JT_NL
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