5
votes
5answers
303 views

Prove limit of $\displaystyle \frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n}$ exists and lies between $0$ and $1$.

Prove limit of $\displaystyle \frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n}$ exists and lies between $0$ and $1$. So far I have $\displaystyle \lim_{n\rightarrow\infty}\sum_{k=1}^n\frac{1}{k+n}=L$ ...
7
votes
2answers
2k views

Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
0
votes
1answer
26 views

Prove that $\sum_{x=0}^{\lambda - 1}(\frac{\lambda^{x+1}}{x!}-\frac{\lambda ^x}{(x-1)!})=\frac{\lambda^\lambda}{(\lambda - 1)!}$

Prove that $\displaystyle\sum_{x=0}^{\lambda - 1}\left(\frac{\lambda^{x+1}}{x!}-\frac{\lambda ^x}{(x-1)!}\right)=\frac{\lambda^\lambda}{(\lambda - 1)!}$. I think I need to write it in the derivative ...
0
votes
2answers
101 views

Solving for a Generating Function in a Special Case

I'm trying to teach myself about generating functions by following this text. I've hit a stumbling block in one of the exercises left for the reader (Sec. 1.4), which I'd quite like to resolve before ...
5
votes
2answers
240 views

Checking of a solution to How to show that $\lim \sup a_nb_n=ab$

In course of solving the problem How to show that $\lim \sup a_nb_n=ab$ I feel that I've probably made some mistake in my solution for I didn't use the fact that $a_n>0$ $\forall$ $n\geq1.$ ...