Consider the series - frac{5}{6} + frac{25}{36} - frac{125}{216} + frac{625}{1296} + ... ...

Question:

Consider the series

{eq}- \frac{5}{6} + \frac{25}{36} - \frac{125}{216} + \frac{625}{1296} + ... {/eq}

Determine whether the series converges, and if it converges, determine its value.

Geometric Series:

Consider a series {eq}\displaystyle t+tb+tb^2+tb^3+tb^4+.............. \infty {/eq}

This is a infinite geometric series whose value is given by {eq}\displaystyle \frac{t}{1-b} {/eq}

The following rules are relevant to this problem:

1.{eq}{{{\left( { - 1} \right)}^{2j}} = 1,\left( - \right).\left( - \right) = + } {/eq}

2.{eq}{1 + \frac{r}{d} = \frac{{d + r}}{d}} {/eq}

Given that: {eq}\displaystyle - \frac{5}{6} + \frac{{25}}{{36}} - \frac{{125}}{{216}} + \frac{{625}}{{1296}} + ......... {/eq}

{eq}\displaystyle \eqalign{ & - \frac{5}{6} + \frac{{25}}{{36}} - \frac{{125}}{{216}} + \frac{{625}}{{1296}} + ......... \cr & \left( { - \frac{5}{6}} \right) + {\left( { - \frac{5}{6}} \right)^2} + {\left( { - \frac{5}{6}} \right)^3} + {\left( { - \frac{5}{6}} \right)^4} + .........\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{{\left( { - 1} \right)}^{2j}} = 1} \right) \cr & \sum\limits_{n = 1}^\infty {{{\left( { - \frac{5}{6}} \right)}^n}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{From series form}}} \right) \cr & \frac{{\left( { - \frac{5}{6}} \right)}}{{1 - \left( { - \frac{5}{6}} \right)}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\sum\limits_{n = 0}^\infty {t{b^n} = \frac{t}{{1 - b}},\left| b \right| < 1} } \right) \cr & \frac{{\left( { - \frac{5}{6}} \right)}}{{1 + \frac{5}{6}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\left( - \right).\left( - \right) = + } \right) \cr & \frac{{\left( { - \frac{5}{6}} \right)}}{{\frac{{6 + 5}}{6}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {1 + \frac{r}{d} = \frac{{d + r}}{d}} \right) \cr & \frac{{ - 5}}{{11}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{Cancel the common term}}} \right) \cr & - \frac{5}{{11}} \cr & \cr & - \frac{5}{6} + \frac{{25}}{{36}} - \frac{{125}}{{216}} + \frac{{625}}{{1296}} + ......... = - \frac{5}{{11}} \cr} {/eq}