I know I have seen something similar and there is a telescoping trick to the convergence but it is eluding me.
4665
Recently, I encountered a problem about infinite series. So my question is how to know whether the infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$ is convergent?
24460
I am trying to mirror an object and the original is symmetrical along all axis, but when I attempt to mirror across the y axis, the mirrored side is tilted to a different angle.
85396
Metric tensor multiplied by its inverse. I always see this with different indices.
21227
Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
68700
I came across this when learning about sequences and series. No one proved it to me tho! Is there a link anywhere? Or would it be beyond what I know? (First term of analysis)
28178
was following a tutorial where I stumbled into this weird line in my mirrored mesh (sorry for the feet lol)      You can also download the blender file
120642
It appeared recently. And I can't even imagine what is it.      Screenshot (click image for larger variant)
94656
In his online lectures on Computational Science, Prof. Gilbert Strang often interprets divergence as the "transpose" of the gradient, for example  (at 32:30), however he does not explain the reason.  How is it that the divergence can be interpreted as the transpose of the gradient?
7595
In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen.   I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's very easy to draw the diagram wrong and have two lines intersect in the wrong place (or intersect when they shouldn't). If someone could find and link this, I would appreciate it very much.
36236
Is there a (real) sequence such that $\sum{a_n}$ diverges but $\sum{na_n}$converges?
82381
Is the following series convergent?  $$\sum_{n=1}^{\infty}\frac{e^n\,n!}{n^n}$$  I treid the Ratio and Root tests, but both of them failed.
108813
I have the following sequences $(a_n)$, $(b_n)=(a_{2n})$, and $(c_n) = (a_{2n-1})$.  I have to show that:   if $(b_n)$ and $(c_n)$ both convergence to $a$, then $(a_n)$ also converges to $a$; if $(a_n)$ converges then $(b_n)$ and $(c_n)$ also converge; there exist $(a_n)$ not converging but with $(b_n)$ and $(c_n)$ converging.   I am not sure how to solve this since I only know that $(b_n)$ > $(c_n)$ > $(a_n)$ on the terms how the sequences are defined.
113699
What does it mean when it looks like this? I'm not sure what I did wrong?
8268
I need to solve    $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$  Does this converge or diverge and why?
52242
I have no idea how to prove it, any advice?
50138
Could someone give me an example of two convergent series $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ such that $\sum_{n=0}^\infty a_nb_n$ diverges?
5433
The problem is the following: suppose $s_1&gt;s_2&gt;0$ and let $s_{n+1}=\frac{1}{2} (s_n+s_{n-1})$ for $n\geq 2$. Show that ($s_n$) converges.  Now, here is what I figured out:   $s_2&lt;s_4$: Base Case for induction that $s_{2n}$ is an increasing sequence. Assume $s_{2n-2}&lt;s_{2n}$. Induction step: $s_{2n}&lt;s_{2n+2}$ $s_1&gt;s_3$: Base Case for induction that $s_{2n-1}$ is a decreasing sequence. Assume $s_{2n-1}&lt;s_{2n-3}$. Induction step: $s_{2n+1}&lt;s_{2n-1}$.   I have proved those two. However arguing in favor of convergence has me going around in circles. Since $s_1&gt;s_2$ and (as discovered during the formulation of Base Cases) $s_3&gt;s_4$, I figured it might be a good idea ot assume that if every odd member of the original sequence ($s_n$) is greater than the following even member, then the limit would be somewhere in between, the two (odd and even) sequences won't cross. Hence the upper and lower bounds would be $s_1$ and $s_2$ respectively. Here is how I approach this:   Assume $s_{2n-1}&gt;s_{2n}$ Show that $s_{2n+1}&gt;s_{2n+2}$   The proof as I mentioned has me running in circles. Any assistance?