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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Prove increasing sequence

I need to prove that $$\left(1-\dfrac{1}{2^n}\right)^n\ge \dfrac12$$ for $n\ge1$. I thought I would prove this by showing that this is an increasing sequence with a base case at $n=1: (1-1/2^1)^1=1/2$, but I can't figure it out. Ideas?
c44
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Convergence of a sequence and its limit

Let $\{v_n\}$ be a sequence defined by $v_1=1$ and $v_{n+1}=\sqrt{v_n^2 +(1/5)^n}$, for $n\ge1$. Then find limit of the sequence $\{v_n\}$. I have found that the given sequence is monotone increasing but failed to find the limit. Please help me to…
Iamdark
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Question about series

how would I find $x$ in the following: $$ 8.056=x+x^2+x^3+\ldots +x^{10}.$$ Well I know the right hand sides simplifies to $$x\cdot \frac{1-x^{10}}{1-x}$$ but I don't know how to proceed further.
Jill
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Evaluate $\sum \frac{2^k}{3^{2^k}+1}$

Evaluate the sum $$\sum_{k=0}^{\infty} \left ( \frac{2^k}{3^{2^k}+1} \right )$$ I first tried to see weather the sum actually converges, by ratio test it does as $\lim_{n\to \infty}|\frac{a_{n+1}}{a_n}|<1$ where $a_n$ is the nth term. Now I am left…
user428700
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does the series $\sum_{n\geq 1}{\frac{(-1)^{\lfloor n-\ln n\rfloor}}{n}}$ converge?

if we consider the series $\sum_{n\geq 1}{\frac{(-1)^{\lfloor| n-\ln n |\rfloor}}{n}}.$ It seems like an example of Leibniz, but here we have ${\lfloor | n-\ln n |\rfloor}$ which is annoying, how can we prove if it is convergent or divergent?
ron macguire
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Convergence comparison test

One of the examples in my textbook uses the comparison $$\frac{1}{\sqrt{k} \ln{k}} \geq \frac{1}{k}$$ This comparison is not immediately obvious to me. I can understand that $\frac{1}{\sqrt{k}} \geq \frac{1}{k}$ holds but I would have assumed that…
Eiraus
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Why does order matter in a conditionally converging series?

Since elementary school, I was taught that addition is unaffected by order. Why does it matter in what order the terms of a conditionally converging series are added? Summation is addition, therefore shouldn't it have the same properties?
sdf
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Please show me how to calculate the folowing limit of a sequence

I think should be something very simple... $$\lim_{n\to\infty}nq^{n}$$ wher $|q|<1$ i tried to use binomial theorem, but no success... I know that $q^{n}$ is convergent to $0$, but can't deal with this $n$
Mykolas
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Is the remainder of the Taylor series of $\exp(x)$ negative when $x<0$ and $n$ is even?

When $x<0$, the following inequality is true: $$e^x<1+x+\frac{x^2}{2!}+\dots+\frac{x^{2n}}{(2n)!}$$ for $n\in\mathbb{N}^+$. My approach: I think it is equivalent to $$\sum_{k=2n+1}^{\infty}\frac{x^k}{k!}<0$$ when $x<0$. If $0>x\geq-(2n+2)$, the…
Laura
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Evalutaing a sum of a cool sequence

So I was messing around, and just thought of a random sequence: $$a_1=3,a_2=5,a_n=a_{n-1}a_{n-2}$$ I wanted to then find the sum of the first $n$ terms of this, and started as so: $$\sum_{k=1}^n a_n = 3+5+3\dot{}5+3\dot{}5^2+3^2\dot{}5^3+... =…
Vedvart1
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Show sequence converge to $0$

Let $(v_k)_{k\geq0}\subset \mathbb{R}_+$ be s sequence such that $v_{k+1}\leq u_kv_k$, where $(u_k)_{k\geq 0} \subset [0,1]$ satisfies $$\sum_{k=0}^{\infty}(1-u_k)=\infty$$ Show that $v_k\rightarrow 0$. Any hints?
user112358
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Finding value of $\lfloor b_{100} \rfloor $ in recursive relation

If $\displaystyle b_{n+1} = b_{n}+\frac{1}{b_{n}}$ and $b_{1} = 1,$ then find the value of $\lfloor b_{100}\rfloor$. My attempt: $\displaystyle b_{n+1}b_{n} = b^2_{n}+1\Rightarrow b_{n}b_{n+1}-b^2_{n} = 1$ $$\frac{1}{b_{n}b_{n+1}-b^2_{n}} =…
DXT
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How to show $\sum_{n=1}^\infty \frac{(-1)^{n}}{n^p},p \geq 1$ converges to a value $\geq -1$

I'm trying to find what $\sum_{n=1}^\infty \frac{(-1)^{n}}{n^p},p\geq1$ converges to. I know from the alternating series test that it converges. I've tried different values of $p$ and Mathematica and found that when p is very large, it converges to…
JoosLover523
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Is there always an answer to "what is the next number in the sequence"

This is my 1st post. I am not a mathematician- so please 'dumb down' any answers :-) I having been thinking about these sorts of "IQ" puzzles: What is the next number in the sequence: $1,2,4,8,16,32, \dots$ In this case, $y=2^x$ If I pick a series…
plant
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How do we show that $S=\sum_{k=1}^{\infty}{k^3\over (4k^2-1)(9k^2-1)(16k^2-1)}?$

Let: $$S=\sum_{k=1}^{\infty}{k^3\over (4k^2-1)(9k^2-1)(16k^2-1)}\tag1$$ How can we show that $$S={1\over 420}+{1\over 7}\ln\left[2^{1/12}\left({4\over 3\sqrt{3}}\right)^{1/5}\right]?\tag2$$ An attempt: Apply decomposition of partial fraction…
gymbvghjkgkjkhgfkl
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