Finding convergence of a series of real exponentiated numbers

Question

I think my teacher should haev written series instead of sequence?

In any case, first off: I just proved that {$x_n$} = $q^n$ converges to $0$ as $n$ goes to $\infty$ for $0<q<1$. So, I at least know that when we have such a q, that my final term $q^n$ in this series goes to zero, and then $q^{n+1}$ . . .

Can I make a sort of induction argument here?

Answer 1

Hint: We have (for $q\ne 0$) $$1+q+q^2+\cdots +q^n=\frac{1-q^{n+1}}{1-q}.$$