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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 .

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proof the empirical distribution converge to distribution in probability

As the image showed, it is a problem of proving the Glivenko-cantelli theorem, but not so complicated. For the distribution F is concentrated on three mass points which means the samples are bounded. And here the empirical distribution converge in…
Yanwen FANG
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Condition for series to converge

If $x_j$ is a positive sequence, what is the condition on $x_j$ so that $$\sum_{j=1}^\infty \exp(-x_j^2)$$ converges? EDIT: A general condition doesn't exist. My question therefore becomes: if $x_j^2\geq j^{1/r}$ for some positive $r$, does the…
sjage
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Is $\int_{0}^{\infty} \log (1+ 2\operatorname{sech}x)\,\mathrm dx$ convergent?

$$\int_{0}^{\infty} \log (1+ 2\operatorname{sech}x)\,\mathrm dx$$ Comparison test isn't helpful in finding the convergence, besides I can't really find the point of discontinuity.
Iti Shree
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Sample maximum estimator

Suppose $x_1, \cdots, x_n$ is a random sample from a uniform $(0, \theta)$ distribution. Suppose $\theta$ is unknown. Let $y_n= \max (x_1,\cdots, x_n)$. Based on the cdf of $y_n$, it is seen that $y_n$ converged in probability to $\theta$,…
L.mak
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For what values of $m$ and $n$ is $\int_{0}^{1} x^{m-1}(1-x)^{n-1}\log x\ dx$ convergent?

I cannot really understand how to prove this I know that $\int_{0}^{1} x^{m-1} (1-x)^{n-1}\ dx$ is $\beta$ function.
Iti Shree
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Prove or provide a counter example about convergent sequence

Prove or provide a counter example: If $a_n \rightarrow 0$ and $\dfrac {a_{n+1}} {a_n} \rightarrow L$ then $L \in [-1;1]$. I think this is right and try to use contradiction to prove it but I still have no idea about this.
Dota2
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If the sequence $\left\{a_n\right\}$ has no two subsequence converging to two different limits then the sequence has a limit?

If the sequence $\left\{a_n\right\}$ has no two subsequence converging to two different limits then the sequence has a limit? Can I understand the question in the way that: If all the convergent subsequences of $\left\{a_n\right\}$ converge to the…
Dota2
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Testing for convergence using Ratio test

The $n$th term of a series is $$\ \frac{n}{\sqrt{n+1}}$$ Upon using the D' Almbert Ratio Test I get $0$ as the limit implying convergence whereas the series seems divergent (each term greater than its respective term of the divergent series…
mathnoob123
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Convergence in distribution implies convergence in probability?

I got confused by a silly question. Suppose $\sqrt nX_n\rightarrow_dN(0,1)$, so $\sqrt nX_n=O_p(1)$, which implies $X_n=O_p(n^{-1/2})=o_p(1)$. Does this mean convergence in distribution implies convergence in probability?
wsong
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Prove that series convergent exactly when another series convergent.

$(a_k)_{k\geq 1} $ is a monotonically decreasing sequence with numbers $ \geq 0$. Prove: The series $\sum_{k=1}^{\infty} a_k $ convergent $\Leftrightarrow$ $\sum_{k=1}^{\infty} 2^k*a_{2^k}$ convergent. My toughts were that if the series on the left…
Razmo
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Convergence integrals equivalent functions.

For, example, question - is $\int_0^{\infty} \frac{x}{x+1}$ convergence Only one place, where we interesting in this integral is ($\infty$). We can take function $g(x)$, that $\lim\limits_{x \rightarrow \infty} \frac{\frac{x}{x + 1}}{g(x)} = n \in…
Eugene Korotkov
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Is $\int_0^{+\infty}x^2 \sin(\frac{\cos x^3}{x+1})$ convergent, absolutely convergent?

Is $\int_0^{+\infty}x^2 \sin(\frac{\cos x^3}{x+1})$ convergent, absolutely convergent? I tried to use equivalence to $\sin(\frac{\cos x^3}{x+1})$ - $\frac{\cos x^3}{x+1}$. But i cannot use this method, cause this function isn't > 0 on all of the…
Eugene Korotkov
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The sequence $(-1)^{n}(1+n^{-1})$ Converges or Diverges?

The sequence $(-1)^{n}(1+n^{-1})$ : a)Converges to $1$. b)Converges to $-1$. c)Converges to $1$ and $-1$ d)Converges to neither $1$ or $-1$. The sequence looks like: $$-1-\frac{1}{1}+1+\frac{1}{2}-1-\frac{1}{3}+1+\frac{1}{4}....$$ $$\Rightarrow…
Shraddha
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$\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\frac{4}{2^{4}}...$ is convergent or divergent?

$$\lim_{n \to \infty } \frac{n}{2^{n}}$$ $$ \text{(using L'Hopitals rule)}$$ $$\lim_{n \to \infty } \frac{1}{n2^{n-1}}$$ $$\frac{1}{\infty }=0$$ $\therefore $ Convergent? Is this the correct solution?
Idkwoman
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Is this series bounded?

$$ \frac{2^n3^n}{n!} $$ I know the series is not monotonoic. I believe the series is bounded below at when n goes to infinity the series will go to 0, but never hit it. I just don't know if the series is bounded above.
Dank memer
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