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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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Summing $3+7+14+24+37...$ up to $10$ terms

What is $3+7+14+24+37...$ up to $10$ terms? I only see that difference between two consecutive terms is in AP ie $7-3=4,14-7=7,24-14=10$ and so on. Any ideas on how to do it?
Archis Welankar
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A series that can be rearranged to converge to any number converges conditionally

Riemann's theorem states that if a series is conditionally convergent, then for any number $L$ (could be infinite), the series can be rearranged in such manner that it would converge to $L$. I was wondering, is the converse true? More formally, let…
Joshhh
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Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$

Prove that $\displaystyle \sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$. I tried using the partial fraction decomposition $a_j = \frac{1}{2j} - \frac{1}{j+1} + \frac{1}{2(j+2)}$, but I don't see how that helps.
Puzzled417
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Sum of $\frac{1}{5}-\frac{2}{5^2}+\frac{3}{5^3}-\frac{4}{5^4}+....$

Find the sum of following series: $S=\frac{1}{5}-\frac{2}{5^2}+\frac{3}{5^3}-\frac{4}{5^4}+....$ upto infinite terms Could someone give me slight hint to solve this question?
Sid1234
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Does a sequence that is eventually constant have less terms than one that is not?

Does a sequence that is eventually constant contain less terms than one that is not? I don't know how to properly think about this, one could either argue that a sequence : 1 2 3 1 1 1 1 1 1 1 1 ..., contains as many terms as 1 2 3 4 5 6 7... or one…
Vlad - Gay Pride World Wide
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About the sequence satisfying $a_n=a_{n-1}a_{n+1}-1$

"Consider sequences of positive real numbers of the form x,2000,y,..., in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 2001 appear somewhere in the…
lest
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write an explicit formula for the sum $\sum_{i=1}^n {(3i+1)}$

I've been shown that : $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ Now I need to write an explicit formula for the sum: $$\sum_{i=1}^n {(3i+1)}$$ I've come up with an answer that is: $$\sum_{i=1}^n {(3i+1)} = \frac{9n^2 + 6n + 3}{2}$$ However I am not…
jksk
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Funny Joke about series

Infinitely many mathematicians walk into a bar. The first says, "I'll have a beer." The second says, "I'll have half a beer." The third says, "I'll have a quarter of a beer." The barman pulls out just two beers. The mathematicians are all like,…
User
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With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges

Let $y_n$ be a sequence of real numbers. Prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges. Let us suppose that $y_{n}$ goes to a limit $L$. Then for all $\varepsilon >0$, for sufficiently large $n$, $|y_n -L| < \varepsilon$…
user321656
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Prove this $|a_{n}|\le 2$

Let sequence such $$|a_{n}-\dfrac{a_{n+1}}{2}|\le 1$$ if $|a_{n}|\le \dfrac{3^n}{2^n}$, show that $$|a_{n}|\le 2$$ since $$2a_{n}-2\le a_{n+1}\le 2a_{n}+2$$ so we $$a_{n+1}-2\ge 2(a_{n}-2)\ge\cdots \ge 2^n(a_{1}-2)$$
math110
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How to prove $|a_{2n}-a_{n}|<\frac{10}{27}$

Let sequence $$a_{1}=1,a_{n+1}=\dfrac{2}{2a_{n}+1}$$ show that $$\left|a_{2n}-a_{n}\right|<\dfrac{10}{27}$$ and the constant $\frac{10}{27}$ A smaller number…
math110
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Convergence of a series

I have to search for convergence of the series $\sum_{n=0}^{\infty} \frac{n+1}{n^2+2n+3}$. I tried to do this by ratio and root test, but unfortunately both are one. I already found out that it is divergent. So I tried to show this by comparison…
SR23
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Sum of the series $\sum \frac{n}{2^{n}}$

I know that the series converges by d'Alembert ratio test, where $\lim\left ( \frac{A_{n+1}}{A_{n}} \right )= \frac{1}{2}$, but I don't know how to calculate the sum of the serie. Thanks for the help.
Iching Sebastian Quares
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What does $\lim_{n \to \infty} \sum_{m=n}^{2n} \frac{1}{m}$ equal?

Possible Duplicate: Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$? I encountered the following sum in Boros & Moll's "Irresistible Integrals" question 5.2.11 (pg. 78): $$\lim_{n \to \infty} \sum_{m=n}^{2n}…
Heebie
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Evaluating a doubly infinite sum

I am evaluating the following infinite sum but I keep getting an answer that is inconsistent with some other result that I know to be true. I would appreciate it if someone could check my solution. Here is the sum I want to…
Calculon
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