# Tagged Questions

37 views

### convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
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### Summing various rearrangements of $1-\frac12+\frac13-\frac14+\cdots$

Show that the series $1-\frac12+\frac13-\frac14+\ldots$ converges to $\log2$ but the rearranged series: ...
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### Convergence of series $\sum$$u_n= \sum$$\frac{n! x^n}{(n+1)^n}$

My series is $$1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots$$ My approach: $$u_n= \frac{n! x^n}{(n+1)^n}$$ So, $$u_{n+1}= \frac{(n+1)! x^{n+1}}{(n+2)^{n+1}}$$ So, ...
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### Test for convergence of the series.

$$\frac{1}{1 \cdot 3}\ + \frac{2}{3 \cdot 5}\ + \frac{3}{5 \cdot 7}\ + \cdots$$ What is the $nth$ term here and what test should I use?
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### Is this positive term series convergent?

My series is: $\frac{1}{1+2^{-1}}\ +\frac{1}{1+2^{-2}}\ +\ldots$ I see my $nth$ term is $\frac{1}{1+2^{-n}}$ How do I test for its convergence?
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### Test for convergence of positive term series

$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$ I can see that the $nth$ term is $\frac{n+1}{n^p}$ How do I test for its convergence?
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### Limit problem using sequential criteria for limits

$$\lim(n+n^2\log \frac{n}{n+1})= \frac12$$ How? In the text book it is simply said that this happens by Sequential criteria of limits. I don't get it.
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### If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.

If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\}$ will converge uniformly to $fg$? I am absolutely stuck on this. Please help.
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### How to show the convergence of this infinite series: $\frac{x}{1+x}- \frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}\dots$

My series is $$\frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-.....$$ Given: $0<x<1$ I see that my nth term is $(-1)^{n+1} (\frac{x^n}{1+x^n})$ My approach was to use Dirichlet's test. ...
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### Prove that the series is convergent: 1- (1+1/3)/2 + (1+1/3+1/5)/3-…

I can see that this is an alternating series with the $n$-th term $$(-1)^{n+1}\frac{1+\frac13+\frac15+\cdots+ \frac{1}{2n-1}}{n}.$$ What test can I apply to show that it converges? Also, it ...
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### rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
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### Analysis Arithmetic series .Verify which of the following sequences converge.

Verify which of the following sequences converge.$$1.A(n)=\sum_{n=1,n=+00}(1/(n^{1+1/n})$$ $$2. B(n)=(1/\sqrt{n^2+1})+.......n/\sqrt{n^2+n}$$ $$3.C(n)=(n+cos(n^2))/(n+sin(n))$$ .For the 3th one ...
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### Does it converges? $\sum \frac{\sqrt{n}\ln(n)}{n^2+1}$

As I checked on Wolfram Alpha I know that $$\sum \frac{\sqrt{n}\ln(n)}{n^2+1}$$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
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### Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
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### Formula of Squaring Sums / Integrals

I'm trying to find a proof for the identities (which I use quite often) $$\left ( \int_{a}^{\infty}f(x)\,dx \right )^2=\int_{a}^{\infty}\int_{a}^{\infty}f(x, y)\,dx\,dy$$ and similarly for the series ...
### Can we expect to find $r,$ large enough, so, $\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C (1+y^{2})^{s}$ for all $y\in \mathbb R$?
Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we expect to find $r$ large enough, so that ...