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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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Showing convergence to $\pm 1$ depending on initial condition?

I'm stuck on this seemingly easy question, hoping someone can explain to me how to prove it. Let $$x_1 = \frac{a^2+b^2+2}{2(a+b)}$$ and for $n > 1$ define $$x_{n+1} = \frac{x_n^2+1}{2x_n}=\frac{x_n}{2}+\frac{1}{2x_n}$$ Then the question asks to show…
potato12
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Convergence in distribution for sequence of Normal RV'S(Compound distribution Gaussian)

let $X_n$ be sequence of random variables with the associated distribution $N(\mu_n,\beta_n+E)$. That is a sequence of normally distributed random variables with changing mean and variance. $\beta_n$ is scalar sequence. $\mu_n$ is a sequence of…
kangkan Dc
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2 answers

Past exam question: Convergence of a series or not

Is the series $$\sum_{k=1}^{\infty} \frac{\sqrt{k}}{k^2+1}$$ convergent? This behaves similar to $\frac{\sqrt{k}}{k^2} = \frac{1}{k^{3/2}}$ so do we use the comparison test???
squenshl
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Convergence of locally integrable functions - L_loc

Given that $f_j, g_j$ and $f,g$ are locally integrable functions, i.e. they are in $L_{loc}^1(\mathbb{R})$. Under the assumption that $f'_j(x)=g_j(x)$ in the sense of distributions, I wanna show that $f'(x)=g(x)$ in $\mathbb{R}$ in the sense of…
Jonathan Kiersch
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What can we say about convergence of $ \frac{\sum_{s=1}^k a_s^2b_s}{\sum_{s=1}^k a_s} $ when $\sum a_n=\infty$ and $(b_n)\geq 0$?

My original question is Assumption on Mirror Descent convergence? but I realized that it boils down into the following question: Suppose $\sum a_n =\infty$ as $n \rightarrow \infty$ where $(a_n)$ is positive and $a_n \rightarrow 0$. Also, $(b_n)$…
user494522
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Interval of convergence 2

How can I find the interval of convergence of: $$\sum_{k=1}^{\infty} \dfrac {1\cdot 3\cdot 5\cdot \ldots \cdot (2k-1)}{2\cdot 5 \cdot 8 \cdot \ldots \cdot (3k-1)}x^k$$ Thought it can be solved with using integral but I couldn't solve.
Person X
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Proving a product of sequences does not converge in measure to 0

I have a candidate counter-example to show that on a set of infinite measure, a product $\{f_{n} g_{n} \}$ of sequences $\{f_{n} \} \to f$ in measure and $\{g_{n} \} \to g$ in measure doesn't necessarily converge in measure to $fg$. Other…
BMac
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Convergence using power-series and convergence-radius

For which values $x \in \mathbb{R}$ does the function $f(x)=k^3*3^{-k}*(x+3)^k$ converge? Use the power-series and convergence-radius approach I have to translate the function into a power series with the format $\sum{a_k x^k}$ I seem to lack the…
blacksmth
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2 answers

Limit of smallest real solution to equation

I think if you solve $$x \sum_{i=0}^{n} (1-x)^i = 0.085$$ for x, the equation always has one real solution such that $0 \leq x \leq 1$, and as $n \rightarrow \infty$, this solution converges towards 0.085 (EDIT: sorry, this was wrong in the original…
Hinton
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For each of the statements below, prove the result if it is true, give a counter example if it is not true

For each of the statements below, prove the result if it is true, give a counter example if it is not true. Suppose $\sum a_n$ with $a_n> 0$ is convergent. Then, a) $\sum \sqrt{a_na_{n+1}}$ is convergent b) for all $0< \delta < 1$, $\sum…
Henam
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If $\sum_{n=1}^\infty x_n$ converges to s, and $y_n = \frac{x_n + x_{n+1}}{2}$ for all $n$, does $\sum_{n=1}^\infty y_n$ converge?

if $\sum_{n=1}^\infty y_n$ does converge, what does it converge to? I have already tried to rearrange $y_n$ but I don't know where to go from here or how the convergence of $x_{n+1}$ can be found.
lionofhk
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Does $ (a_{n}) = \sqrt{(4- (b_{n}) ^2} $, where $ (b_{n}) =\frac{1}{2}( 1+ \frac{1}{n}) ^{n} $ converge towards $ \sqrt{4- (e/2) ^2} $?

Does $$ (a_{n}) = \sqrt{(4- (b_{n}) ^2} $$, where $$ (b_{n}) =\frac{1}{2}( 1+ \frac{1}{n}) ^{n} $$ converge towards $ \sqrt{4- (e/2) ^2} $? Because $(b_{n})$ converges towards $\frac{e}{2}$?
user15269
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2 answers

convert recursive to implicit $X_k = (X_{k-1} * 1.4) - 230$

I've got a small recursive formula: $X_k = (X_{k-1} * 1.4) - 230$ How can I convert this to a implicit formula? Best regards
Lukas
  • 103
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1 answer

How to calculate the sum of $\sum\limits_{n=1}^{\infty}\left(\frac{4^{n-1}}{5^{n-1}}+\frac{4}{5^{n-1}}\right)$

I know this series is converges by limit test, but i can't find a way to calculate its sum. $$\sum\limits_{n=1}^{\infty}\left(\frac{4^{n-1}}{5^{n-1}}+\frac{4}{5^{n-1}}\right)$$ Thanks for helping,Here is the solution. Solutions…
PatricEz
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5 answers

How do I determine the divergence/convergence of $\sum_n \frac{1}{\log(\log(n))}$?

I am working through some problems in Durrett's probability book, and one of them involves a variant of the law of iterated logarithm. I've managed to reduce the result to showing that $$\sum_n \frac{1}{\log \log (n)}\exp(-\log \log(n)) <…
Xiaomi
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