Ratio Test — Question 3

Question 3 – Solution

Step 1: Compute the exact ratio.

$$\frac{a_{n+1}}{a_n}=\frac{(n+1)^2}{2^{n+1}}\frac{2^n}{n^2}=\frac12\left(1+\frac1n\right)^2.$$

Step 2: Evaluate the limit.

$\displaystyle L=\frac12(1)^2=\frac12<1$.

Conclusion.

The series converges absolutely. Its consecutive terms eventually behave as though multiplied by $1/2$. The polynomial factor changes the ratio by a factor tending to $1$, while $2^n$ supplies the decisive factor $1/2$.