Root Test — Question 3

*Question 3 Find the radius and interval of convergence of $$\sum_{n=1}^{\infty}(3^n+2^n)^{1/n}x^n.$$ Use the Root Test for the interior and test both endpoints separately.

*Question 3 – Solution

For $a_n=(3^n+2^n)^{1/n}x^n$, $$\sqrt[n]{|a_n|}=(3^n+2^n)^{1/n^2}|x|.$$ Write $3^n+2^n=3^n[1+(2/3)^n]$. Then $$(3^n+2^n)^{1/n^2}=3^{1/n}\left[1+\left(\frac23\right)^n\right]^{1/n^2}\longrightarrow1.$$ Thus the Root Test limit is $L=|x|$: convergence holds for $|x|<1$ and divergence for $|x|>1$.

Since $(3^n+2^n)^{1/n}\to3$, the terms at $x=1$ approach $3$, and at $x=-1$ their magnitudes approach $3$. Neither endpoint passes the nth-term test.

The radius is $\boxed{R=1}$ and the interval is $\boxed{(-1,1)}$.