Calc1.8

From Example Problems
Jump to: navigation, search

Investigate the convergence of this series: 3-2+{\frac  {4}{3}}-{\frac  {8}{9}}+...+3\left(-\,{\frac  {2}{3}}\right)^{k}+...\,

Rewrite the series.

=3\left[1-{\frac  {2}{3}}+{\frac  {4}{9}}-{\frac  {8}{27}}+...+\left(-\,{\frac  {2}{3}}\right)^{k}+...\right]\,

=3\sum _{{k=0}}^{\infty }\left(-\,{\frac  {2}{3}}\right)^{k}\,

This is a geometric series with r=-2/3\,, so it converges to 3\left({\frac  {1}{1+2/3}}\right)={\frac  {9}{5}}\,.

Calculus

Main Page