Root Test — Question 2

*Question 2 Analyze $$\sum_{n=1}^{\infty}\left(1+\frac{(-1)^n}{2}\right)^n.$$

1. Examine the even and odd terms separately.

2. Apply the nth-term divergence test.

3. Confirm the conclusion using the limsup form of the Root Test.

*Question 2 – Solution

If $n=2k$, then $$a_{2k}=\left(1+\frac12\right)^{2k}=\left(\frac32\right)^{2k}\longrightarrow\infty.$$ If $n=2k+1$, then $a_{2k+1}=(1/2)^{2k+1}\to0$.

A convergent series must have $a_n\to0$. The even subsequence is unbounded, so the full term sequence does not approach zero. Hence the series diverges.

$$\sqrt[n]{|a_n|}=1+\frac{(-1)^n}{2},\qquad \limsup_{n\to\infty}\sqrt[n]{|a_n|}=\frac32>1.$$ The limsup Root Test gives the same result; an ordinary root limit does not exist.