Q 49. - Calculus

Question

Determine whether the series $\sum _{j=2}^{\infty }\frac{2^j}{3}$ converges or diverges. Give the sum of the convergent series.

Step-by-Step Solution

Answer

The series $\sum _{j=2}^{\infty }\frac{2^j}{3}$ diverges.

Step 1. Given information.

Given a series $\sum _{j=2}^{\infty }\frac{2^j}{3}$.

Step 2. Find if the series converges or not.

The index starts with 2, rather than 0.

Note that the convergence of a series depends not upon the first few terms but only upon the tail of the series.

The standard form of a geometric series is $\sum _{k=0}^{\infty }c{r}^k$.

The geometric series converges if and only if $\left|r\right|<1$.

Here, the series $\sum _{j=2}^{\infty }\frac{2^j}{3}$ has $c=\frac{1}{3}$ and $r=2$.

Since $r=2$, it follows that the series $\sum _{j=2}^{\infty }\frac{2^j}{3}$ diverges.