Part 1: For what values of $k$ is the limit $<1$:
$\lim_{n\to \infty} \dfrac{(n+1)^4}{kn +k!} < 1$ where we can choose $k$ to satisfy this inequality.
How would I start? I see where $k=n$ the limit is clearly $0$ but what else can I do?
Part 2: Is this ratio test correct: $\sum \limits_1^\infty \dfrac{(n!)^4}{(kn)!}$
Compute:
$\lim_{n \to \infty} \dfrac{((n+1)!)^4}{(k(n+1))!} \cdot \dfrac{(kn)!}{(n!)^4} = \dfrac{(n+1)^4(n!)^4}{(kn +k)(kn+k-1)(kn+k-2)...(kn)!} \cdot \dfrac{(kn)!}{(n!)^4}$
Which is:
$\dfrac{(n+1)^4}{(kn)^k +something + k!}$ So I would say the series converges when $k \geq 4$
- Is it possible to easily determine "something" or would this be your analysis?