Evaluate the convergence of the following infinite sum. I am so bad when it comes to parameters. I don't know where to start. $\sum_{n=1}^{\infty}\dfrac{(n+1)^a-n^a}{n^b}$, where $a,b\in\mathbb{R}$.
123411
I was wondering how the convergence of the series $$\sum_{k\geq1}\frac{(k+1)^y-k^y}{k^x},$$ depends on $x$ and $y$, where $x$ and $y$ are real numbers. I've been able to show that it converges if and only if $y\leq x$, by distinguishing nine cases; whether $x$ and $x-y$ are positive, negative or zero. If desired I can include a sketch of my proof. My question is whether there is a proof that works for all real $x$ and $y$? Or at least a proof that seems less artificial?
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?
77115
Let $a_n=1-2^{\frac{-1}{n}}$, $b_n=\frac{1}{n^p}$. Does $\sum_{n=1}^{\infty}a_n$ converge or diverge when compared to $\sum_{n=1}^{\infty}b_n$ if $p &lt; 1$? My answer to this is that it diverges, since $b_n$ diverges if $p&lt;1$, and by a comparison test $a_n$ does as well. Is this correct? Thank you.
123540
My teacher posed an infinite series question to me today and I'm not quite sure how to start to go about it. $$\sum_{r=1}^{\infty} \dfrac{2^{-r}}{r^{2}}$$ Any hints would be much appreciated.
15494
Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is.
52391
The question is to calculate the radius of convergence of $\sum\limits_{n=0}^\infty n!x^{n^2}$.  I dont know how to calculate R.O.C for such type of questions involving $n^2$ as power of x. How do we do this? Any help appreciated.
92841
convergence of $ \sum_{k=1}^\infty \sin^2(\frac 1 k)$   How can I do this? Should I use the Ratio Test (I tried this but it started getting complicated so I stopped)? Or the Comparison test(what should I compare it to?)?
65900
Please explain different methods to calculate the sum of infinite series with $\dfrac{n^2}{2^n}$ as it's general term i.e. Calculate   $$\sum_{n=0}^\infty \dfrac {n^2}{2^n}$$  Please avoid the method used for summation of arithmetic geometric series. It is very tedious approach. Does any other simpler approach exist?
47091
My question is pretty simple.  Since $n! \gt n!!$, it's clear by the comparison test that $\sum_{n=0}^\infty \frac {1}{n!!}$ converges.      But what value does the sum converge to? How does one go about determining its value (if possible)?
54332
I have  two series  $\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$   $\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$  with radius of convergence $2$ and $3$ respectively.  How can I find the radius of convergence for the following?     $\displaystyle\sum_{n=1}^{\infty} (a_n +b_n)x^{n}$   Can anyone help me?
24190
I got this question in my maths paper     Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$   and find the sum if it exists.   I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you.
119835
I'm trying to prove the convergence of $$ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}$$ with $\alpha &gt; 1$.   For $\alpha \geq 2$ I can use the comparison test ($\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges) so I'm missing $2&gt;\alpha&gt;1$ and I'm pretty much out of ideas.  If you could offer some advice I would very much appreciate it. Thanks in advance
74315
What is the radius of convergence of this power series here?   $$\sum_{n=1}^{\infty}2^n x^{n^2}$$
36066
So I need to know the convergence of this sum. The integral test does not seem to work (WolframAlpha gives a confusing answer which implies it's not the correct method). I also tried the limit comparison test for various harmonic series, but they are all inconclusive. I'm also not getting very far with the ratio test. $$ \sum_{n=2}^\infty \dfrac{\sqrt{n+1}}{n(n-1)} $$
20864
Is $$\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$$ uniformly convergent on $(0,+\infty)$? Sadly, Weierstrass' criterion nor Cauchy's condition don't seem to work here and I just ran out of ideas.
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 .
40706
Let $a_n$ be a sequence. $\lim_{n\to\infty} a_n =0 $ then the series: $\sum_{n=1}^\infty a_n, \ \sum_{n=1}^\infty (a_n+a_{n+1}) $ converge and diverge together.  It doesn't seem right, there are no convergence tests that fit this question yet every counterexample I try does not seem to work. Is this how I should see the sum? $\sum_{n=1}^\infty (a_n+a_{n+1})=2\sum_{n=1}^\infty a_n +a_{n+1}$ If so then there's no real difference between the two series since you multiply it by a constant and add just one more value.