Root Test — Question 10

*Question 10 Let $c\in\mathbb R$. Determine exactly which values of $c$ make $$\sum_{n=1}^{\infty}c^{n^2}$$ converge. Apply the Root Test where decisive and inspect the boundary cases separately.

*Question 10 – Solution

$$\sqrt[n]{|c^{n^2}|}=|c|^n.$$

• If $|c|<1$, then $|c|^n\to0<1$, so the series converges absolutely.

• If $|c|>1$, then $|c|^n\to\infty$, and the terms $c^{n^2}$ do not approach zero.

• If $|c|=1$, the Root Test gives the boundary value $1$. Directly, $|c^{n^2}|=1$, so the terms again fail to approach zero.

The series converges exactly when $\boxed{|c|<1}$, equivalently $\boxed{-1<c<1}$. Both endpoints diverge.