Root Test — Question 10

Question 10 – Solution

Step 1: Compute the root expression.

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

Step 2: Separate the parameter cases.

• 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.

Conclusion.

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