question determine if the series -3 - 9/2 - 27/4 - 81/8 - 243/16 - 729/32 - 2187/64 + ... converges or…

question determine if the series -3 - 9/2 - 27/4 - 81/8 - 243/16 - 729/32 - 2187/64 + ... converges or diverges. if the series converges, find the sum. answer attempt 1 out of 2 -3 - 9/2 - 27/4 - 81/8 - 243/16 - 729/32 - 2187/64 + ... geometric series.

question determine if the series -3 - 9/2 - 27/4 - 81/8 - 243/16 - 729/32 - 2187/64 + ... converges or diverges. if the series converges, find the sum. answer attempt 1 out of 2 -3 - 9/2 - 27/4 - 81/8 - 243/16 - 729/32 - 2187/64 + ... geometric series.

Answer

Explanation:

Step1: Identify the first - term and common ratio

The first - term $a=-3$. To find the common ratio $r$, divide the second term by the first term. $\frac{-\frac{9}{2}}{-3}=\frac{3}{2}$.

Step2: Apply the convergence test for geometric series

For a geometric series $\sum_{n = 0}^{\infty}ar^{n}$, it converges if $|r|\lt1$ and diverges if $|r|\geq1$. Here, $|r|=\left|\frac{3}{2}\right|=\frac{3}{2}\gt1$.

Answer:

The series diverges.