Problem 1 Evaluate
∞
Σ ( 100^n (2n)! / (3n)! )
n=1
Converges or diverges?
And,
Problem 2 Evaluate
∞
Σ [(n^3 + n)/(2n^3 + 5n^2 − 3n + 1)]^(n+1)
n=1
Converges or diverges?
I get the following using the ratio test:
$\displaystyle \lim_{n \to \infty} \Big| \frac {100^{n+1}(2n+2)!}{(3n+3)!} \frac {(3n)!}{(2n)!100^{n}} \Big| $ $\displaystyle = \lim_{n \to \infty} \frac {100(2n+2)(2n+1)}{(3n+3)(3n+2)(3n+1)} = 0 $ since the the numerator is a quadratic and the denominator is a cubic
$\displaystyle lim_{n\rightarrow\infty}\left(\frac{100^{n+1}(2n+2 )!}{(3n+3)!}\frac{(3n)!}{(2n)!100^{n}}\right)$
$\displaystyle lim_{n\rightarrow\infty}\left(\frac{100^{n+1}(2n+2 )(2n+1)(2n)!}{(3n+3)(3n+2)(3n+1)(3n)!}\frac{(3n)!} {(2n)!100^{n}}\right)$
as you can see (2n)! divide (2n)! =1 the same thing to (3n)!