Power Series — Question 2

Question 2 – Solution

Step 1: Use the Ratio Test.

For $a_n=nx^n/4^n$, $$\left|\frac{a_{n+1}}{a_n}\right|=\frac{n+1}{n}\frac{|x|}{4}\longrightarrow\frac{|x|}{4}.$$ Convergence requires $|x|/4<1$, so $|x|<4$ and $R=4$.

Step 2: Test the endpoints.

• At $x=4$, the terms are $n$, which do not approach $0$.

• At $x=-4$, the terms are $n(-1)^n$, whose magnitudes grow.

Both endpoint series diverge by the nth-term test.

Conclusion.

The interval is $\boxed{(-4,4)}$. The factor $(n+1)/n\to1$, so $n$ does not alter the radius; at the endpoints it prevents the terms from tending to zero.