Proving summation series inequality

[karan000]

Question
http://puu.sh/52zAa.png

Attempt
http://puu.sh/52AVq.png

I've attempted to use Riemann sums and use the integral to prove the inequality, not sure if this was the right approach to start with as I am now stuck and don't see what to do next.

For part (b), I know that if (2√n -2) → ∞ as n → ∞, then Sn → ∞ for n → ∞ hence the summation series is divergent.

 

[mfb]

$S_n > 2 \sqrt{n}-2$ and $S_n > \sqrt{n}$ are not the same.
I think you are supposed to use induction in (a). The integral approach works, but it needs more mathematics.

For part (b), I know that if (2√n -2) → ∞ as n → ∞, then Sn → ∞ for n → ∞ hence the summation series is divergent.

Good, as (2√n -2) → ∞ for n → ∞ is true.

 

[mathnerd15]

does it help to shift the series from k=1 to k=2 which differ only by 1? if you can prove the inequality holds for Sn>2sqrt(n)>sqrt(n) since n>0

maybe there are cases where you can shift the sum by a real number so that the first term of the sum is equal to k in the integral 2sqrt(n)-k ? Euler proved the series convergence for k=1 to n=infinity, 1/n^2=pi^2/6