Question

Determine whether the series is absolutely convergent, conditionally convergent,or divergent.

\(\sum\limits_{k = 1}^\infty {k*{{(\frac{2}{3})}^k}} \)

Short answer

The Series is Absolutely Convergent.

Step-by-step solution

(Using Ratio Test)

\({\lim _{n \to \infty }}|\frac{{{a_{n + 1}}}}{{{a_n}}}|\)=L

Then Series is Absolutely Convergent if\(L < 1\),

And Divergent When\(L > 1\)or\(\infty \)\(\)

(Check For Convergence..)

\({\lim _{n \to \infty }}|\frac{{{a_{n + 1}}}}{{{a_n}}}|\)\( = {\lim _{n \to \infty }}|{\frac{{(k + 1)(\frac{2}{3})}}{{k{{(\frac{2}{3})}^k}}}^{k + 1}}|\)

=\({\lim _{n \to \infty }}\frac{2}{3}|\frac{{k + 1}}{k}|\)

=\(\begin{aligned}\frac{2}{3}\\\end{aligned}\)

=\(\frac{2}{3} < 1\)

Hence,By Ratio Test The Series is Convergent..

\(\)Therefore From the Results,The given Series is Absolutely Convergent..