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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Write down an example of a sequence ${a_n}$ such that ${a_n}$ is divergent, $a_n < 5$ for all $n$, and $a_n < a_{n+1}$ for all $n$

Write down an example of a sequence ${a_n}$ such that: ${a_n}$ is divergent $a_n < 5$ for all $n$, and $a_n < a_{n+1}$ for all $n$, or explain why no such divergent sequence exists. What I did thus far: We know that the sequence is increasing since…
Cherry_Developer
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Prove equality $\sqrt{1+x^2}\cdot\ln (x+\sqrt{1+x^2}) = x+\frac{x^3}{3} - \frac{2}{3}\frac{x^5}{5}+ \frac{2}{3}\frac{4}{5}\frac{x^7}{7}-...$

$\sqrt{1+x^2}\cdot\ln (x+\sqrt{1+x^2}) = x+\frac{x^3}{3} - \frac{2}{3}\frac{x^5}{5}+ \frac{2}{3}\frac{4}{5}\frac{x^7}{7}-...$ I stuck with performing right part. Where are these coefficients from?
Desh
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A dubious proof using Weierstrass-M test for $\sum^n_{k=1}\frac{x^k}{k}$

I have been trying to prove the uniform convergence of the series $$f_{n}(x)=\sum^n_{k=1}\frac{x^k}{k}$$ Obviously, the series converges only for $x\in(-1,1)$. Consequently, I decided to split this into two intervals: $(-1,0]$ and $[0,1)$ and see…
Johnny Westerling
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Sum of infinite series $\cos \theta - \frac{\cos 5\theta}{5} + \frac{\cos 7\theta}{7} - \frac{\cos 11\theta}{11} ..., \theta \in (-\pi/3, \pi/3)$

I'm stumped at the following exercise on series summation: $$\cos \theta - \frac{\cos 5\theta}{5} + \frac{\cos 7\theta}{7} - \frac{\cos 11\theta}{11} ..., \theta \in (-\pi/3, \pi/3)$$ The range gave me a hint that I should (probably?) split this…
Jimmie Lin
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Binomial expansion for approximation

I've just had to do a homework on binomial expansion for approximation: $1.07^9$ so: $(1+0.07)^9$ To do binomial expansion you need a calculator for the combinations button (nCr), so why would use a more complicated method, which only gives an…
Jonathan.
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How to calculate infinite series of complex numbers?

I have to solve these problems and show how to solve them in front of students tomorrow though I haven't learned the way to calculate infinite series of complex numbers. (1) $\sum _{n=0} ^{\infty} z^n$ (2) $\sum _{n=0} ^{\infty} (3^{n+1} - 2^{n+2})…
ynn
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Find the sum $\sum_{n=1}^{\infty} \frac{4n}{n^4+2n^2+9}$

Find the sum $$\sum_{n=1}^{\infty} \dfrac{4n}{n^4+2n^2+9}.$$ By calculator, we can predict that its sum is equal to $\dfrac{5}{6}$ so I think we should use inequalities to prove it. And I found that $\dfrac{5}{6(n^4+n^2)} < \dfrac{4n}{n^4+2n^2+9}<…
liverpool29
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Inequality of Finite Harmonic Series

I'm asked to prove that for $n\in \mathbb{N}$ $$\frac{1}{1} + \frac{1}{2} +\cdots+\frac{1}{n} \geq 1 + \frac{n}{2}$$ by induction. I've got a feeling that the problem isn't right (since it isn't true for any $n\in \mathbb{N}$), does anyone know the…
htd
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Find the sum $\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{99\cdot 100}$

Please help me calculate the following sum $$\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{99\cdot 100}$$
user39471
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Fubini theorem for sequences

I want to find a counter example This is the Fubini theorem for sequences: If $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}|a_{mn}|<\infty,$$ then…
Park
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Can you help me understand this definition for the limit of a sequence?

I'm reading the textbook "Calculus - Early Transcendentals" by Jon Rogawski for my Calculus III university course. I'm trying for the life of me to understand the wording of this definition, and I wonder if it can be said in simpler terms to get the…
Mirrana
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How to calculate an expression based on given information

How to solve the question: If $A=(2+1)(2^2+1)(2^3+1)......(2^{2048}+1)$, what shall be the value of $(A+1)^{1/2048}$. I have not been able to think of anything in any direction, Tobe honest!
asabhish
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How can we show that $\ln{2}=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n}\left({12\over e^{n\pi}-1}+{4\over e^{n\pi}+1}\right)$

$$\ln{2}=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n}\left({12\over e^{n\pi}-1}+{4\over e^{n\pi}+1}\right)\tag1$$ Any hints?
user339807
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Sequence of real numbers.

Let $\{x_{n}\}$ be a sequence of real numbers. Suppose for each $\epsilon>0$, there is a sub sequence $\{x_{n_{k}}\}$ so that $x_{n_{k}}\leq x+\epsilon,k\geq1.$ Then we have $1. \limsup_{n \to \infty}x_{n} \leq x.$ $2. \limsup_{n \to \infty}x_{n}…
neelkanth
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How can we show that $1-\gamma=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n(n+1)}\sum_{k=1}^{\infty}{1\over (k+1)k^n}$

$$1-\gamma=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n(n+1)}\sum_{k=1}^{\infty}{1\over (k+1)k^n}$$ How would we go about proving this series?
user339807
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