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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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Unboundedness of $\{y_n\}$.

$\{x_n \}$ is a positive sequence defined for $n=0,1,2, \cdots $ and satisfies $x_n\geqq \dfrac{1}{2}(x_{n-1}+x_{n+1})$ for $n\geqq 1 \cdots ☆.$ Define $\{y_n\}_{n=1}^{\infty}\ $ by $y_n=x_n-x_{n-1}$. Then, prove that $\{y_n\}$ is a bounded…
daㅤ
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Why doesn't this sequence have any pointwise subsequence?

Prove that the sequence of functions $f_n(x) = \sin(n x)$ have no pointwise convergent subsequence I am confused. If I let $x = 1$, then we get $f_n(1) = \sin(n)$. By Bolzano Weistress, we have a convergent subsequence no?
Lemon
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Uncountable convergent sum of real numbers with uncountably many non-zero terms

Does anyone know of an example of an uncountable convergent sum $\sum_{s \in S} x_s$ of real numbers where uncountably many $x_s$ are non-zero? Ideally I'd like to be able to compute $\sum_{s \in S}x_s$ as well. I know that an uncountable convergent…
Austin Shiner
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Does the square root of a sum of squares grow more slowly than the sum of terms?

Suppose I have a non-negative sequence $a_n \geq 0$ satisfying $\sum a_n = \infty$. I’m trying to show that $\dfrac{\displaystyle \sum a_n}{\sqrt{\displaystyle \sum a_n^2}}=\infty$ Is this always true? It seems to be true for $a_n = 1/n$ and $a_n =…
Asterix
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About the convergence of $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$

Does the series converge? $$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$$
Crazy_girl
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(easy) rearranging of power series denominator

My teacher has done this: $$\frac{1}{z^3(1-z^2/3+O(z^4))} = \frac{1+z^2/3+O(z^4)}{z^3}$$ How does that work? I don't understand why he can claim this.
A. Top
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What should be the value of $\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$?

$\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$ Obviously, by using common-sense the answer is $0$. But I had thought of a different mathematical approach. $Let:$ $x = \sqrt{0+\color{red}{\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0…
Saikat Das
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If $\sum a_n=0$, what can we say about $\sum n a_n$?

Suppose we know that $$\sum_{n=0}^{k}(-1)^na_n=0$$ for even $k$, and that $\{a_n\}$ is decreasing after $n/2$. What conditions can we set on $a_n$ to make $$\sum_{n=0}^k(-1)^na_n(-n)$$ either positive or negative?
user81654
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Convergence of the Series - $\sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n}{(3k-1)}}{\prod_{k=1}^{n}(4k-3)}$

Prove that the following series is convergent. $$\sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n}{(3k-1)}}{\prod_{k=1}^{n}(4k-3)}$$ I don't know for where to begin.
Otávio Rapôso
  • 570
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About series representations

Can $\displaystyle \frac{1}{1+\frac{z}{n}}$ have the following series representation? $\displaystyle \sum_{k=0}^{\infty}\frac{(-z)^k}{n^k}$
Bon Les
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Maclaurin series for functions which are not infinitely differentiable

I was introduced to Maclaurin series through $\sin(x)$, $\cos(x)$ and $e^x$. I have always thought that Maclaurin series works for these functions because they are infinitely differentiable. My question is; Does this also work for functions which…
DeriveMe
  • 91
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Stupid question about $1 - \frac{1}{2}-\frac{1}{4}+\frac{1}{3}- \frac{1}{6}-\frac{1}{8}+\frac{1}{5}\dots$

I have $$1 - \frac{1}{2}-\frac{1}{4}+\frac{1}{3}- \frac{1}{6}-\frac{1}{8}+\frac{1}{5}\dots$$ Partial sum $S_{3n}$ of the above is: $$(1 - \frac{1}{2}-\frac{1}{4})+(\frac{1}{3}- \frac{1}{6}-\frac{1}{8})+(\frac{1}{5}-\dots$$ But what is $S_{3n-1}$ and…
17SI.34SA
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Guess the production rule for this sequence

In this question, I’ll temporarily bypass MSE rules and wait a little bit before revealing the context, because I believe people looking at it with a fresh eye will be able to see several things that I'm unable to see now. I’ve computed the six…
Ewan Delanoy
  • 61,600
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Relation between these numbers

Can someone please tell me the relation between these numbers it can be of any type e.g they have common difference or something like that The numbers are - $80327$, $803267$, $8032673$, $80327000$, $803267063$
Kartik Khare
  • 91
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Converge of this series

I was shown this example, but do not understand the procedure for doing this and why it's correct. Show $\sum x^{n!}$ converges for $x \in (0,1)$. Way it was shown: Write this as $x+x^2+0x^3+0x^4+\dots$.i.e. $\sum c_kx^k$ where $c_k$ is $1$ if…
Scott Frazier
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