How to calculate limit of series

Question

I have many limits for homework that I dont know how to solve them.

I tried many things, but dont have any idea. Hope you can help me

$$\lim_{n\to \infty} n*c^n $$ when $$\lvert c\rvert < 1$$

one more limit is: $$\lim_{n\to \infty} \frac{\sqrt[n]e + \sqrt[n] {e^2} + \sqrt[n]{e^3}+...+\sqrt[n]{e^{2n}}}{n}$$

Thanks a lot.

Answer 1

Hint: Take the logarithm of $nc^n$: $$ \log(n)+n\log(c)=n\,\overbrace{\left(\frac{\log(n)}{n}+\log(c)\right)}^{\text{eventually }\le\frac12\log(c)\,\lt\,0} $$ Use L'Hôpital to find $$ \lim_{n\to\infty}\frac{\log(n)}{n} $$

Answer 2

HINT:

For the first problem, since $c<1$, define a number $x>0$ by $c=\frac{1}{1+x}$. Then,

$$nc^n=\frac{n}{(1+x)^n}$$

Now expand the numerator using the binomial theorem.

For the second problem, recognize the sum is the Riemann sum for $\int_0^2 e^x\,dx$.