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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What is the telescoping series? $\sum_{k=1}^{n} k \cdot k!$

So we learned telescoping series in class and I came across this question in my textbook and I tried to evaluate it, but I don't understand how to do it. $$ \sum\limits_{k=1}^n\left(k \cdot k!\right) $$ According to the answers: $$…
user2351149
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Find the sum of series without differentiation

Given a series $\sum_{i > 0}\frac{i^2}{z^i}$, and $\sum_{i > 0}\frac{i}{z^i} = \frac{z}{(z - 1)^2}$ I need to find the sum My method does not require differentiation but there is a difficulty. Let $S = \frac{1^2}{z} + \frac{2^2}{z^2} +…
errorcodemonkey
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sum of terms of series

If $$F(t)=\displaystyle\sum_{n=1}^t\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$$ find $F(60)$. I tried manipulating the general term(of sequence) in the form $V(n)-V(n-1)$ to calculate the sum by cancellation but went nowhere. I also tried…
Albus Dumbledore
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Infinite Summation of positive numbers is? Infinite?

I was thinking the following: A sum of an infinite number of values, no matter how big, but positive equals infinite. And then I discovered the following row: $\sum_{n=0}^\infty (\frac{1}{2^{n}})$ This is: $ 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +…
Martijn Courteaux
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How Can I prove $\sum_{n=1}^{\infty}{\frac{n^4}{5^n}}=\frac{285}{128}$

Question:Prove that $ \sum_{n=1}^{\infty}{\frac{n^4}{5^n}}=\frac{285}{128}$ While doing questions on series and products,i got stuck in this question.I was not able to figure out any way how to prove this one.Convergence tests shows that this series…
Paras
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find the minimum of the value $|a_{1}+a_{2}+\cdots+a_{11}|$

let sequence $a_{n}$ such $a_{1}=0$,and such $$|a_{n+1}|=|a_{n}+1|,\forall n\in N^{+}$$ find the minimum of the value ( ) $$f=|a_{1}+a_{2}+\cdots+a_{11}|$$ $A:3$ $~~~~~~~~$ $B:2$ $~~~~~~~~$ $C: 1 $ $~~~~~~~~$ $D:0$ I try: since…
math110
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Question about proof the harmonic series diverges

The usual proof that the harmonic series diverges that I have seen involves group together terms that sum to $\frac{1}{2}$. It may take the form: $$\sum\limits_{n=1}^{\infty} \frac{1}{n} \ge 1 + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right)…
John P.
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Limit of $\frac{1}{n}\sum\limits_{i =2}^n \frac{1}{\ln i}$ as $n \to \infty$

For the sequence $a_n =\frac{1}{n}\sum_{i =2}^n \frac{1}{\ln i}$ (with $n \ge 2$), I would like to determine if the limit exists, and, if so, find its value. Some observations I have made so far: integral comparison does not seem to help--we do have…
JZS
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Does there exist a sequence $\{a_n\}_{n \ge 0}$ of nonnegative reals such that $ \sum_{j \ge 0} a_{nj} = \frac{1}{n}$ holds for all naturals $n$?

Does there exist a sequence $\{a_n\}_{n \ge 0}$ of nonnegative reals such that $$ \sum_{j \ge 0} a_{nj} = \dfrac{1}{n}$$ holds for all naturals $n$? My progress: I could show that $a_n\le \frac{1}{2n}$. I am not sure if this is even useful.
Takamoto Yuji
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A product sequence

Evaluate $$\frac{\prod_{i=1}^n[(2i-1)^4+ 1/4]}{\prod_{i=1}^n[(2i)^4+ 1/4]}$$ First I thought I would multiply both the numerator and the denominator by the denominator itself. Now, I am unable to evaluate the series. I would appreciate innovative…
vaibhav
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How to find the sum of a convergent series

I am given the following geometric series and am asked to find the sum. $$\sum_{n=1}^{\infty} \left(\frac{12}{(-5)^n}\right)$$ I know that I somehow need to get this in the form $\sum_{n=1}^{\infty}ar^{n-1}$, where $a$ is the first term and $r$ is…
dtg
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$a_{n+1}a_n=a_n^2+a_n+1$, $a_1=1$

$a_{n+1}a_n=a_n^2+a_n+1$, $a_1=1$,how to find integer $k$, make $\left|\sqrt{a_{2020}}-k\right|$ as small as possible. Using computer, I get $a_{2020}\approx 2027.38$. then $\sqrt{a_{2020}}\approx 45.0264,k=45$ But without computer, how to calculate…
AsukaMinato
  • 1,007
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Does there exist an explicit expression for the series ( or function) $f(x)=\sum \limits_{n=1}^\infty e^{-xn^2}$?

Does there exist an explicit expression for the series (or function) $$f(x)=\sum _{n=1}^\infty e^{-xn^2}\text{ ?}$$
user71282
  • 41
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Is a sequence involving multiplication of the previous 2 numbers' non-zero digits eventually periodic?

Assume that the first two numbers of a sequence are natural numbers $n_1$ and $n_2$ (they may be equal). Every following number in the sequence is calculated by multiplying all non-zero digits of the previous two numbers. Is this sequence eventually…
haal_
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What is the name of this series? (or type of series?)

Math newbie here! The picture above shows my "working out" (or attempt rather). And I've managed to nail down the right name for one of the sums (labeled 1) as a sum of triangle numbers! But I can't figure out what the second sums is? (if it's…
yiffyiffyiff
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