Solve the problem.
-If $\sum a _ { n }$ is a convergent series of nonnegative terms, what can be said about $\sum \sqrt [ k ] { a _ { n } }$, where $k$ is a positive integer?
A) May converge or diverge
B) Always converges
C) Always diverges

Question

Quizplus · Accepted Answer

By applying the root test, we have 
\begin{align*}
\lim_{n	o\infty}\sqrt[k]{a_n} &= \left(\lim_{n	o\infty}a_night)^{1/k}\
&=0^{1/k}\
&=0.
\end{align*}
Thus, the convergence of $\sum\sqrt[k]{a_n}$ is determined by the convergence of the geometric series 
$$\sum_{n=1}^\infty \left(\sqrt[k]{a_n}ight)^{k}=\sum_{n=1}^\infty a_n.$$ 
Since $\sum a_n$ is convergent, $\sum\sqrt[k]{a_n}$ may converge or diverge, depending on the behavior of the terms.

Solve the Problem ∑an\sum a _ { n }∑an​ Is a Convergent Series of Nonnegative Terms, What Can Be

Solve the Problem $\sum a _ { n }$ Is a Convergent Series of Nonnegative Terms, What Can Be