Question

How to find the sum of the infinite geometric series given \[\dfrac{5}{3} - \dfrac{{10}}{9} + \dfrac{{20}}{{27}} - ...?\]

Answer 1

Hint: For finding the summation of infinite geometric series you should use certain formulas which include the formulae of summation of infinite series. For solving such series you should know the first term and the common factor (ratio of the second term to the first term), since the last term is not known in such questions you have to deal with these two data only.

Complete step by step answer:
Using formula of summation \[S = \dfrac{{{a_1}}}{{1 - r}}\]
Here, S= summation of the series, a1= first term, r= common ratio.
Now finding a common ratio by taking the ratio of the first and second term
We get \[\dfrac{{ - 10}}{9} \div \dfrac{5}{3} = \dfrac{{ - 2}}{3}\]
Now using formulae we get,
\[
  S = (\dfrac{5}{3}) \div (1 - \dfrac{{ - 2}}{3}) \\
  S = \,25 \\
 \]
So a summation of series is \[25\].
Formulae Used: Summation of infinite geometric series is used.
Additional Information: You can take care that the ratio lies between \[ - 1 < r < 1\], if not then there is some fault either in question or in your calculation, or the series is not geometric.

Note:
 Geometric series is the series in which the common ratio i.e. the ratio of the second term to the first term or the third term to the second term, simply the ratio of exceeding term to its previous term should be the same in every case. So once you recognize the series then the solution process is just you have to take care of formulas and have to apply them carefully. It’s not a very big deal in solving the geometric series but the thing is sometime recognition of the series become so tough by just seeing it, so in that case, you have to take ratios and then you can easily find it