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Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.

Formally, a sequence $S_n$ converges to the limit $S.$

$$\lim_{n\to \infty}S_n=S $$ if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.

A sequence diverges if it does not converge.

For more specialized questions about divergent series, such as summation methods, consider the tag .

21990 questions
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Can a function be square integrable without being integrable?

Reading Tolstov's 'Fourier Series', which states that $f(x)$ is square integrable if both $f$ and its square both have finite integrals over some interval. I haven't seen this restriction on $f$ before, which makes me wonder - can squaring a…
Benjamin Lindqvist
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9
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3 answers

Proving that unconditional convergence is equivalent to absolute convergence

Regarding this discussion here: Absolute convergence a criterion for unconditional convergence. (thank you for the great answers, by the way) I'm still trying to do Exercise 3.2.2 (b) from these notes by Dr. Pete Clark. The question states that for…
roo
  • 5,598
8
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3 answers

Disproving uniform convergence

Given a sequence of functions $f_n$ which is known to be pointwise convergent how would you go about showing that is not uniform convergent? The particular example I'm working with is $f_n (x)=\frac {nx} {x^2+n^2}$ and I've tried using Theorem…
Fabien dB
  • 81
8
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1 answer

Does $\sum\int\arctan(n^2x)\sin(1/x)\,dx$ converge?

Does the series $$\sum_{n = 1}^\infty\int_0^{\frac{1}{n}} \arctan(n^2 x)\sin\left(\frac{1}{x}\right)dx$$ converge? I have tried to estimate this integral, but I can't get upper estimate better than $\frac{1}{n}$. Can you give me a hint, please?
ErlGrey
  • 333
6
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1 answer

Weak Convergence in $L^p$

Suppose $p\geq 2$ and $\Omega\subset\mathbb{R}^n$ is a bounded domain. Suppose that $u_n\in L^p (\Omega)$ and $u_n\rightharpoonup u$. is true that $u_n^2\rightharpoonup u^2$ in $L^\frac{p}{2}(\Omega)$? Thanks
Tomás
  • 22,559
6
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1 answer

Does this intergal $\int_0^\infty\frac{ \sin^2 x} x\,dx$ converge?

I'm going to make use of the series $\displaystyle \sum_{n=0}^\infty \frac 1{n+1}$. and that $\displaystyle \int_0^\infty \frac{ \sin^2 x} x \, dx = \sum_{n=0}^\infty\int_{n\pi}^{(n+1)\pi} \frac{ \sin^2 x} x \,dx$ If I use variable substitution…
stuck
  • 91
6
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1 answer

Convergence of Series of a Net's terms

I'm working through Dr. Pete Clark's convergence notes here: http://alpha.math.uga.edu/~pete/convergence.pdf and I've been thinking about Exercise 3.2.2 (a) and I am completely stumped. The exercise says to show that a series converges only if it…
roo
  • 5,598
5
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2 answers

usage of this condition

A very widely stated result: A sequence $x_n \to x$ iff every subsequence $x_{n^{\prime}}$ of $x_n$ contains a further subsequence $x_{n^{\prime\prime}}$ such that $x_{n^{\prime\prime}}\to x$. My question is that there is no need to pass on to…
user24367
  • 1,286
5
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3 answers

If $a_k>0$ and $\sum_{n=1}^{\infty} a_k$ converges then there exists $\gamma_n\to \infty$ such that $\sum_{n=1}^{\infty} \gamma_ka_k$ converges

Question: Suppose $a_k>0$ for $k\in\mathbb{N}$ and the series $\sum_{k=1}^{\infty} a_k$ converges. Prove that there exists a sequence $(\gamma_k)_{k=1}^{\infty}$ with $\gamma_k>0$ and $\gamma_k\to \infty$ when $n\to \infty$ such that…
lecdabster
  • 205
5
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0 answers

What conditions for $f(x)$ does $\lim_{x\to\infty} f(x)-\sum_{n=1}^{x}f'(n)$ converge

I was exploring the euler-mascheroni constant when I thought of this problem. The euler-mascheroni constant (usually denoted as $\gamma$) can be calculated a number of ways, but the primary example is this formula:…
ThePretzelMan
  • 352
5
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2 answers

convergence with comparison test

$$\sum_{n=1}^\infty \frac{4n}{n^4+2n+9}$$ Hi guys! I need to use comparison test on this series, I haven't done a lot of comparison tests so far, so I'm not sure what to compare it with. Should I use $$\frac{4n}{n^4}$$ which is $$\frac{4}{n^3},$$…
Stefana
  • 71
5
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1 answer

Is the convergence linear, superlinear, quadratic?

The Fibonacci sequence is generated by the formulas \begin{cases} r_0=1 & r_1=1\\ r_{n+1}=r_n+r_{n-1} \end{cases} The sequence therefore starts out $1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$. Prove that the sequence $[2r_n/r_{n-1}]$ converges…
Username Unknown
  • 4,633
5
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1 answer

Does removing all numbers in the harmonic series with a units digit of 9 affect the series?

The harmonic sequence is $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}\ldots$ diverges. There is a simple reason why: $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}\ldots$ is obviously greater…
ETS1331
  • 1,335
5
votes
2 answers

Prove that if the absolute value of a sequence converges to $0$, the sequence converges to $0$ as well.

Let ($a_n)_{n \in \mathbb{N}}$ be a sequence, prove that if $|a_n|$ converges to $0$ then ($a_n)_{n \in \mathbb{N}}$ converges to 0 as well. Now let $|a_n|$ converge to $0$ and let $\epsilon > 0$ that means that $||a_n|-0| < \epsilon$ for an $N \in…
JonDoeMaths
  • 145
5
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1 answer

The Comparison test or The limit comparison test

I have checked many of this site's pages yet I could not find a clear answer about how to choose between the "comparison test" OR the "limit comparison test". Because the difference between the two is that in the limit comparison test, one has to…
aziz
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