Evaluate the two limits.
$$ \lim_{n\to \infty} \left(\frac1n\right)^{15}\sum_{k=1}^nk^{15} \tag{1} $$ $$ \lim_{n\to \infty} \left(\frac1n\right)^{17}\sum_{k=1}^nk^{15} \tag{2}$$
can anyone please help me with them? I know that I should use Riemann sum, but I'm not sure how or what function's Riemann integral looks this way.
Consider $$\frac{1}{n}\sum_1^n \left(\frac{k}{n}\right)^{15}.\tag{1}$$ This is a right Riemann sum for $$\int_0^1 x^{15}\,dx.$$ The limit as $n\to\infty$ of (1) exists. From that, you should be able to find the answers to both questions.
Remark: We used a Riemann sum, since that seemed to be the approach requested. Another way of viewing things is that $\sum_1^n k^{15}$ is a polynomial in $n$ of degree $16$. Thus the first sequence obviously diverges to $\infty$, and the second converges to $0$.