question determine if the series 256/27 + 1024/81 + 4096/243 + 16384/729 + ... converges or diverges. if the…

Explanation:

Step1: Identify the first - term and common ratio

The first - term $a$ of the geometric series is $a=\frac{256}{27}$. To find the common ratio $r$, divide the second term by the first term. The second term is $\frac{1024}{81}$ and the first term is $\frac{256}{27}$. So, $r = \frac{\frac{1024}{81}}{\frac{256}{27}}=\frac{1024}{81}\times\frac{27}{256}=\frac{4}{3}$.

Step2: Check for convergence

For a geometric series $\sum_{n = 0}^{\infty}ar^{n}$, it converges if $|r|\lt1$ and diverges if $|r|\geq1$. Since $|r|=\left|\frac{4}{3}\right|=\frac{4}{3}> 1$.

Answer:

The series diverges.