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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Calculating $e$

If I calculate $e$ using the following formula. $$e = \sum_{k=0}^{\infty}{\frac{1}{k!}}$$ Is it possible to predict how many correct decimal places I get when I stop summing at $n$ terms?
iblue
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Convergence of a monotonic sequence

Assume $a_n\geq 0$ is a sequence of positive real numbers which satisfy the following inequality: for each $n,m\in\mathbb{N}$, we have $$(n+m)a_{n+m}\leq na_n+ma_m.$$ I can't show the convergence of this seemingly well-behaved sequence (I am…
Simulacra
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Proving that $a_n=\frac{n^2+2n+6}{n^3-3}\to 0$ as $n\to\infty$

$$a_n=\frac{n^2+2n+6}{n^3-3}$$ So I want to show that "$a_n\to a\iff\forall \epsilon>0,\quad\exists N\in\mathbb{N}:n\geq N\implies |a_n-a|<\epsilon$" Then my rough working: $|a_n-0| =\left|\frac{n^2+2n+6}{n^3-3}\right|<\epsilon$ Estimate…
mrnovice
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Find the sum of the infinite series $\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot4\cdot5\cdot6}+...$

Find the sum of the series $\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot4\cdot5\cdot6}+...$. This type of questions generally require a trick or something and i am not able to figure that out. My…
idpd15
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Calculus II series

I'm trying to find convergence or divergence of $$\sum_{n=1}^\infty\frac{\cos(\log n)}{n}\text{.}$$I've tried to use the squeeze theorem, however, my professor said that $\log(n)$ grows too slowly for the numerator to be assumed to be zero. I don't…
Justin Clark
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Let us define a series $\{a(n)\}$ such that $a(n)= a(n-1) + \frac{1}{a(n-1)}$

Let us define a series $\{a(n)\}$ such that $a(n)= a(n-1) + \frac{1}{a(n-1)}$ and $a(1)=1$ then prove that $a(75)$ belongs to the interval $(12,15)$ note here i have used $X(n)$ where n is in sub script denoting the series count. Im unable to…
Abhash Jha
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Prove the series $\sum_{n=1}^{\infty}{x^{n-1}\over(n-1)!}\cdot{e^{-xn}-1\over e^{xn}-1}=-e^{-x(1-e^{-x})}$

Show that, $$\sum_{n=1}^{\infty}{x^{n-1}\over(n-1)!}\cdot{e^{-xn}-1\over e^{xn}-1}=-e^{-x(1-e^{-x})}$$ My try: We know $$\sum_{n=1}^{\infty}{x^{n-1}\over (n-1)!}=e^x$$ $$\sum_{n=1}^{\infty}{e^{-xn}-1\over e^{xn}-1}={1\over 1-e^{x}}$$ How do I use…
gymbvghjkgkjkhgfkl
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Partial limits of sequence

Problem: Given ${a_n}$ and that 2008 and 2009 are partial limits of ${a_n}$. Also, for all $n$, $|a_{n+1} - a_n| \le 1/2$. Prove that ${a_n}$ has at least 3 partial limits. I attempted to prove it using the definition of partial limits of a sequence…
Ma.H
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Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.

Problem. Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $. I don’t know how to solve this problem... What I can think of is to just simplify this inequality: $$ \left[ \frac{(1 + a) (1 + b) (1 +…
oshhh
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Prove that $n^n$ is greater than $1\cdot3\cdot5\cdots(2n-1)$

Prove that $\ n^n \ge (1)(3)(5)\cdots(2n-1)$ I can't think of how to start answering this question and it would be great help if someone could explain how I should go about doing it. Note:It is a sequence and series question(AP,GP,HP)
oshhh
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Are all infinite sums not divergent? In quantum field theory

I am a physicist interested in physics. In particular this question is related to quantum field theory. I recently came across a derivation of the infinite sum $1+1+1+1+..... $ that produced the result -1/2, aka zeta regularization (from Terry…
user122066
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For what values for m does $\sum \limits_{k=2}^{\infty}\frac{1}{(\ln{k})^m}$ converges?

For what values for m does $$\sum \limits_{k=2}^{\infty}\frac{1}{(\ln{k})^m}$$ converge? What about $$\sum_{k=2}^{\infty}\frac{1}{(\ln(\ln{k}))^m}$$ or more generally $$\sum_{k=2}^{\infty}\frac{1}{(\ln(\cdots (\ln{k}))\cdots)^m}$$ ?
jimjim
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What is a mathematical expression for the sequence $\{1,1,-1,-1,1,1,-1,-1,\dots\}$?

What is a mathematical expression for the sequence $\{1,1,-1,-1,1,1,-1,-1,\dots\}$, that is $1$ and $-1$, two at a time alternating?
Postskjerm
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What is the value of　$\frac 1 {0! + 1! + 2!} +\frac 1 {1! + 2! + 3!} + \frac 1 { 2! + 3! + 4!} + …?$

What is the value of x? $$e = 1 / 0! + 1 / 1! + 1 / 2! + … .$$ $$1 = 1 / (0! + 1!) + 1 / (1! + 2!) + 1 / (2! + 3!) + … .$$ $$(x = ) 1 / (0! + 1! + 2!) + 1 / (1! + 2! + 3!) + 1 / (2! + 3! + 4!) + … .$$ In my calculation by programming, x is about…
TOM
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Proof that a sequence is bounded

I would like to show that the following sequence is bounded: $$c_{n+1}=c_{n-1}+\frac{c_n}{2^{\frac{n}{2}-1}},$$ with $c_0=1$, $c_1=2^{3/4}$. I don't think the initial values are important in any way; I mention them for completeness. I am also…
the_fox
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