Why does the $\lim \sum{n^{(1/n)}-1}$ diverge as $n\rightarrow \infty$? I suspected it would converge as $\lim {n^{(1/n)}}=1$ as $n\rightarrow \infty$ but computation show otherwise. So, now I am trying to show the partial sums are unbounded but didnt get far.
123075
Let $a_n=\sqrt[n]n-1$. Does $\sum_{n=1}^\infty a_n$ converge?
21239
Does the series $\sum \limits _{n=0}^{\infty} \cos(n\pi)$ converge or diverge?   On substituting values I get alternate $1$ and $-1$.    So taking sum of infinite GP, I get $\dfrac 1 2$. So it looks like it converges.
72368
Is the sequence $\Big(\dfrac {\tan n}{n}\Big) $ convergent ? If not convergent , is it properly divergent i.e. tends to   either $+\infty$ or $-\infty$ ? ( Owing to $\tan (n+1)= \dfrac {\tan n + \tan 1}{1- \tan1 \tan n}$ and the non-covergence of   $\Big (\tan n \Big)$ it is easy to see that if $\Big(\dfrac {\tan n}{n}\Big) $ is convergent then the limit must be  $0$ , but I can not proceed further . )
69751
Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to compute any kind of "lim sup" fails. I would appreciate your help in this.
123584
Sorry for the oversimplified question, but does the series $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge? The ratio test of it gives the result of "1". Thanks a lot.
2903
Just thinking about it in terms of logic, shouldn't the series of a sequence whose limit as $n$ approaches infinity is 0 converge?  I know that the $n$th term test for divergence says that if a series is convergent, then the limit of its sequence is 0 and I also know there are some sequences for which it has been "proven" that their series does not converge even though the sequence converges to 0, but I just don't believe these tests. If we stretch $n$ out to infinity and the terms are approaching 0, then how is it possible for the sum of the terms to "overflow" and diverge if the terms are becoming negligibly small?
96153
Let $a_n$ be a positive sequence such that $S_n = \sum\limits_{k=1}^n a_k$ diverges.   I'm trying to prove $\sum\limits_{k=1}^n \frac{a_n}{S_n}$ diverges.   I tried summation by parts, limit comparison and Stolz theorem in many different combinations and am still stuck with nothing.
73242
I am investigating the convergence of  $$\sum _{n=1}^{\infty }\left\{ 1-n\log \frac {2n+1} {2n-1}\right\} $$ Now as per Cauchy's test for absolute convergence.  If $\overline {\lim _\limits{n\rightarrow \infty }}\left| u_{n}\right|^{{1}/{n}} &lt; 1,\sum _\limits{n=1}^{\infty }u_{n}$ converges absolutely  Obviously, if $\overline {\lim \limits_{n\rightarrow \infty }}\left| u_{n}\right|^{{1}/{n}} &gt; 1,\sum _\limits{n=1}^{\infty }u_{n}$ does not converge.  I observed $$\overline {\lim _{n\rightarrow \infty }}\left| \log \left( \dfrac {2n+1} {2n-1}\right) ^{-n}\right|  = \overline {\lim _{n\rightarrow \infty }}\left| \log \left( 1-\dfrac {1} {n-{1}/{2}}\right) ^{-n}\right|$$  Could I take the negative power of $n$ outside the absolute brackets here? I guess even if I could establish $\log$ part converges that would only show that the overall series diverges right. Is that the correct result ? Any alternative lines of attacking this problem would be much appreciated.
68121
Does the following series converge or diverge?  $\sum_{n=1}^{\infty} a_n$ where  $a_n = \frac 1 b_n$, and $(b_n)_n$ is the subsequence of $(n)_n$ whose terms do not have a $9$ in their decimal notation.  I have no idea how the sequence $b_n$ even looks like. If I have an idea of how to prove that a series converges or not with criteria but I don't if I don't know what $b_n$ is.
79190
How would you show that $\lim_{n→∞}\ln(1/n)=−∞$ using the definition of what it means to diverge to $−∞$. If I plug in different $n$ I see that it goes to $−∞$, but I don't understand how to show this. Could someone please demonstrate?
48067
I can clearly see that $\dfrac{n!}{n^n}\to 0$ when $n\to\infty$. But how do I know if the sum $$\sum_{n=1}^{\infty}\frac{n!}{n^n}$$ is convergent or not? I know this might be basic, but thank you if anyone can help me.
8268
I need to solve    $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$  Does this converge or diverge and why?
108931
I've been reading proofs of this theorem, and I was wondering why the following is true:  We know that $\sum\limits_{n=1}^{\infty}\frac{1}{n}$ diverges.  I'm not sure how knowing this fact leads us to the next step:  For any $M &gt;0$ we can find primes $p_1,\cdots p_m$ such that $\left(\sum\limits_{i=1}^{N}\frac{1}{p_1^i}\right)\cdots \left(\sum\limits_{i=1}^{N}\frac{1}{p_m^i}\right)&gt;M.$  What I am saying is, given the opportunity, I would have never thought to look at a finite product of partial sums.  I understand why everything works after the above step.
46652
Why does the series $\sum_{n=1}^∞ \ln ({n \over n+1})$ diverges? I'm looking for an answer using the comparison test, I'm just not sure what I can compare it to.   And can I have some tips on what to look at when handling with series that have logarithms in the expression?   Thanks in advance!
107818
I'm having a lot of trouble figuring this one out.  Determine if  $\sum_{i=2}^{\infty} \frac{(-1)^n}{n+(-1)^n}$ converges or diverges  Both ratio and root test are inconclusive and I'm at a loss. Can anyone help me?
76391
If $\{x_n\}$ is a sequence which satisfies $\lim_{n \rightarrow\infty} ~ x_{n+1} - x_n= c$  where $c$ is a real positive number. Then what can be said about the convergence/ divergence, boundedness/ unboundedness of $\{x_n/n\}$.  Attempt: $\lim_{n \rightarrow\infty} ~ x_{n+1} - x_n= c$ where $c &gt;0$  => $x_n$ is unbounded and divergent.  However, I am stuck on how to relate this to convergence/divergence of $x_n/n$ . Thanks for the help.
33274
The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction?  Thanks