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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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Checking for convergence/divergence of the nth root.

I've been working on this problem for a while, but I can't really get it. I get it, but I don't actually get it. The question is to find whether or not this series converges: $\displaystyle\sum_{n=1}^\infty(2 ^{1/n} - 1) $ I am almost certain that…
Zein
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Show that the hyperplane $H=\{x \in\mathbf{R} \;|\; : {A}^{T} x =b\}$ is closed in $\mathbf{R}^d$.

Let $A \in \mathbf{R}^d \backslash \{0\}$ and $b\in\mathbf{R}$. Show that the hyperplane $H=\{x \in\mathbf{R} \;|\; : {A}^{T} x =b\}$ is closed in $\mathbf{R}^d$.
user187039
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Convergence of two nested sequences

For the two following sequences I want to find their limits: (1) The sequence $2$, $2\sqrt{2}$,$2\sqrt{2\sqrt{2}}$,... (2) $a_{n+1}$ = $\sqrt{1+a_n}$, $a_1 = 1$ For both sequences I want to show that they are bounded and monotone increasing. My…
user66280
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Convergence of $\sum_{n = 0}^{\infty} \frac{qz^n}{1 - qz^n} $

I need to prove that the following series converges $$\sum_{n = 0}^{\infty} \frac{qz^n}{1 - qz^n}$$ for $|q| < 1$, $|z| < 1$. I thought maybe to use some comparison tests, but I'm not sure if that will work. Also, note that I have to prove…
MT_
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Convergence of square of monotone sequence implies convergence of sequence.

Suppose $(a_n)$ is a monotone sequence. Prove that if $(a_n^2)$ is convergent, then so is $(a_n)$. How do I use the monotone convergence theorem to prove this?
user178358
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Convergence of sequences and limits

Let $(f_n)$ be the fibonacci sequence and let $x_n = \dfrac{f_{n+1}}{f_n}$. Given that $\lim_{n \to \infty}(x_n) = L$ exists, determine the value of $L$.
priya
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Sequence of Functions that does not Converge

I'm asked to show that the sequence of functions $f_n(x) = n^2x^n$ defined on the closed interval $[0,1]$ does not converge pointwise to any function as $n \to \infty$. For $0 \le x \lt 1$ I think I have convergence to zero, but for $x = 1$ the…
user164587
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Can someone explain to me why this series diverges?

I have this series $$\sum_{k=1}^{\infty}\frac{1}{k^{3+\cos k}}$$ I understand that if the exponent is fixed (not a function) and greater than 1 the series converges (p-series) but I can see in wolfram that this series diverges clearly (wolfram says…
Masacroso
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Limit of a (pseudo) non-increasing sequence

Consider a non-negative sequence $t_1,t_2,...$ that is also bounded above? Suppose that the sequence is "pseudo non-increasing" in the sense that $t_{n+1} \leq t_n + e_n$ where $e_1 + e_2 + ...$ is finite. Is the sequence necessarily convergent? I…
user171375
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if $\dot{x}<0$ and $\dot{|x|}<0$, what can we say about the convergence of $x$?

we have $\dot{x}=\frac{dx}{dt}<0$ and $\dot{|x|}=\frac{d|x|}{dt}<0$, that is: the derivative of $x$ is negative, the derivative of the absolute value of $x$ is negative, and $t$ is time. what can we say about the convergence of $x$? (which type of…
Martial
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Does $\sum_0^\infty(\frac{1}{9n+1})$ converges?

Does $\sum_0^\infty(\frac{1}{9n+1})$ converges? If yes, then to what?
Satvik Mashkaria
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Convergence of a series $\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$

$$\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$$ I know this series does not converge. Can someone show me how to prove that? Should i use criteria of Dalamber or any other criteria?
user144463
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Show uniform convergence of series

Show that $\sum_{j=0}^{\infty} (j+1)x^j$ converges uniformly for x $\in$ any compact subset of (-1,1). Using the ratio test, I got: $\frac{j+2}{j+1}x^j$. However, I don't know how to compare this to 1...Is there another method, or am I missing…
kiwifruit
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Ratio test and radius of convergence

I need to find the radius of convergence for: $$\sum \ln j^3 x^j$$ By the ratio test, I get: $$\displaystyle\frac{\ln (j+1)^3 x^{j+1}}{\ln j^3x^j}$$ However, I'm not sure what happens to the ln parts in terms of convergence?
kiwifruit
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Convergence of $\int_{0}^{\pi} \frac{\sin{(x)}}{(x+n\pi)^{p}} dx$

$$\int_{0}^{\pi} \frac{\sin{(x)}}{(x+n\pi)^{p}} dx$$ where $p>0$ And $n\in\mathbb{N}$. I understand we can compare this to $$\int_{0}^{\pi} \frac{1}{(x+n\pi)^{p}} dx$$ which tells us it converges for $p\ge1$ but we don't know what happens for $p<1$.…
user2850514
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