Root Test — Question 8

*Question 8 Analyze $$\sum_{n=1}^{\infty}\left(\frac{2+\sin n}{4}\right)^n.$$ Use the limsup form of the Root Test to handle the oscillating base, and explain why an ordinary root limit is unavailable.

*Question 8 – Solution

Since $1\le2+\sin n\le3$, $$\sqrt[n]{|a_n|}=\frac{2+\sin n}{4}\in\left[\frac14,\frac34\right].$$ This sequence oscillates rather than approaching one value.

Integer angles modulo $2\pi$ are dense because $1/(2\pi)$ is irrational. Consequently, $\sin n$ comes arbitrarily close to $1$, so $$L=\limsup_{n\to\infty}\frac{2+\sin n}{4}=\frac34<1.$$ (For convergence alone, the uniform bound $\sqrt[n]{a_n}\le3/4$ already suffices.)

The series converges absolutely by the limsup Root Test.