Evaluate $\lim_{n\rightarrow\infty} (\sum_{k=1}^{n} (\int_{0}^{\frac{π}{2}}(\sin(x))^{k}dx))$

Question

I have to evaluate the following limit of a sum: $\lim_{n\rightarrow\infty} (\sum_{k=1}^{n} (\int_{0}^{\frac{π}{2}}(\sin(x))^{k}dx))$. I ended in $\lim_{n\rightarrow\infty} (\int_{0}^{\frac{π}{2}}(\sum_{k=1}^{n} \frac{(\sin(x))^{n}-sin)}{\sin(x)-1})dx$. From here on however I am unsure if I can introduce the limit under the integral or how to continue.