Suppose $x_1 ≥ x_2 ≥ · · · ≥ 0$. Then $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{n=1}^{\infty} 2^{n}x_{2^n}$ converges. I mean it is evident that $x_{2^n}$ is meant to be a sub sequence of $x_n$ but I'm quite unsure how to proceed.
122575
The following is Cauchy condensation test:  If $b_n \ge 0$ and $b_n \ge b_{n+1}$ then $\sum b_n$ converges if and only if $\sum 2^n b_{2^n}$ converges.  Is the following modified Cauchy condensation test possible:  Let $N \in \mathbb N$. Then if $b_n \ge 0$ and $b_n \ge b_{n+1}$ then $\sum b_n$ converges if and only if $\sum N^n b_{N^n}$ converges?
44043
We let a>0 and b>0 and  we have $\sum_{n=0}^{\infty}b^n-a^n$. We have to find all values a,b>0 where $\sum_{n=0}^{\infty}b^n-a^n$ is convergent. I think I can use that $$\sum_{n=0}^{\infty} b_n &lt; \infty , \sum_{n=0}^{\infty} a_n &lt; \infty \implies \sum_{n=0}^{\infty} (b_n-a_n) &lt; \infty. $$. And I think $\sum_{n=0}^{\infty}b^n$ and $\sum_{n=0}^{\infty}a^n$ is convergent if a&lt;1 and b&lt;1? It will make sense. But how can I show it?
27488
Given a positive sequence $(a_n)$. Prove that if $\displaystyle \sum_{n=1}^{\infty} a_n$ converges, then so does $\displaystyle \sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$.   I tried to use the root test, and I was stuck with the case $\lim \sup \sqrt[n]{a_n}=1$. Maybe we need some trick here.  Thanks so much.
23237
Let $(a_n)_n$ be a sequence of non-negative real numbers.      Prove that $\sum\limits_{n=1}^{\infty}a_n$ converges if and only if the infinite product $\prod\limits_{n=1}^{\infty}(1+a_n)$ converges.   I don't necessarily need a full solution, just a clue on how to proceed would be appreciated.  Could any convergence test like the ratio or the root test help?
30633
Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?  The ratio test is inconclusive.  It passes the term test for $\Re(n)=1$, but this is not sufficient to prove convergence.  Since we are dealing with so many complex numbers, I do not know of any convergence tests for this.  I know that if it is convergent, it is conditionally convergent.
109414
If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges?  I want to use this fact in another problem but don't immediately see how to prove it.
32839
Let's suppose that $(a_n)$ is a sequence so that $\sum a_n^2 n^2$ converges, so I have to prove that $\sum |a_n|$ converges.    By the Cauchy criterion we have that, since $\sum a_n^2 n^2 $ converges if $\varepsilon &gt; 0$ then there is a $k\in\mathbb N$ so for all $m,n$ naturals that satisfy $m &gt; n \geq k$: $$ |n^2 a_n^2 + ... + m^2 a_m^2| &lt; \varepsilon $$ but I can't get $| |a_n| + ... + |a_m|  | &lt; \varepsilon$ to show that $\sum |a_n|$ converges. Could you give me a hint? Is there any way to prove this without resorting Cauchy criterion. Thanks!
6372
If $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}$$ Converges( s is real) and $\operatorname{Re}(z)&gt;s$.  Then $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{z}}$$ also converges.  $a_n$ is complex sequence.
84143
Consider the series $ \sum_{n=1}^\infty x^2+ n/n^2$  . Pick out the true statements: (a) The series converges for all real values of $x$. (b) The series converges uniformly on $\mathbb{R}$. (c) The series does not converge absolutely for any real value of $x$.        stuck on this problem totally. please help me to solve this problem.thanks .
104695
Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also convergent and its sum equals $\alpha \cdot \sum_{n=0} ^{\infty} x_n + \beta \cdot \sum_{n=0} ^{\infty} y_n$?
38720
Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$.  I showed that this series point wise.  Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |x|^n$$ converges, By abosultely convergese test $\sum (-1)f_n(x)$ converges.  I want to show that this series uniformly converges on $[0,1]$.
22436
Let $\sum_{n=1}^{\infty}a^2_n$ and $\sum_{n=1}^{\infty}b^2_n$ prove that: $\sum_{n=1}^{\infty}a_nb_n$ converges.  I thought about this answer: In order for a sum to converge i has to be bounded, Means Let $M_1\epsilon R$, So that $ \sum_{n=1}^{\infty}a_n&lt;M_1 $. Same goes for $b_n$. [And lets say $M_2$].  So is it just: $\sum_{n=1}^{\infty}a_nb_n\le \max(M_1*M_2, \sqrt{M_1*M_2}$)?  And one more question: Let $\sum_{n=1}^{\infty}a^2_n$ prove that  $\sum_{n=1}^{\infty}\frac{a_n}{n}$ converges.  If i go the same as before, I have hard time cause i get something like:  $\sum_{n=1}^{\infty}\frac{M_1 }{n}$ which doesn't converge [I think].  Any guidelines please?
40393
If $\sum a_n$ with $a_n&gt;0$ is convergent, then is $\sum {a_n}^2$ always convergent? Either prove it or give a counter example.  Im trying in this way, Suppose $a_n \in [0,1] \ \forall\  n.\ $ Then ${a_n}^2\leq a_n\ \forall\  n.$ Therefore by comparison test $\sum {a_n}^2$ converges.  So If $a_n$ has certain restrictions then the result is true.  what about the general case?  How to proceed further? Hints will be greatly appreciated.
83427
I have been trying to solve this problem for awhile, and I can't seem to solve it.Below is the question. I suspect that the series does converge, but I am unable to prove this. Thanks for any help in advance.  Let $(a_n)_{n=1}^\infty$ be a convergent sequence of real numbers. Can it be concluded that $\sum\limits_{n=1}^{\infty}|a_{n+1} - a_{n}|$ converges? Give a proof or a counterexample.
71569
So one of my professors proposed a problem to me and it has stumped me for some time now. Here's how it goes:  Suppose you have a sequence $a_n$ of real numbers such that $$\lim_{n\to\infty} a_{n} = 0$$ and suppose the sequence of partial sums $s_n$ is bounded. Prove that $s_n$ converges or give a counterexample.  I'm hoping to figure this out without anyone handing me the complete solution, so if someone could point me in the right direction with a hint, it would be much appreciated.
104844
Let $y_n$ be a sequence of real numbers. Prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges.  Let us suppose that $y_{n}$ goes to a limit $L$. Then for all $\varepsilon &gt;0$, for sufficiently large $n$, $|y_n -L| &lt; \varepsilon$ .  But $|y_n - L| &lt; 2|x_n-\frac{L}{3}|+|x_n - \frac{L}{3}|$  But then how to proceed?
23900
Per the title, if $\sum_{1}^{\infty}(a_n)^3$ diverges, does this imply that $\sum_{1}^{\infty}(a_n)$ diverges?   I'd appreciate hints (!) for dealing with this excercise.  EDIT Per the contrapositive, it is not given that $a_n$ converges absolutely, or that it is nonnegative for all $n$.  Thank you!