Question

How do you find the sum of the infinite geometric series given `1-0.5+0.25-...`?

Answer 1

Sum of the infinite geometric series is `2/3`.

Explanation:

As `-0.5/1=0.25/-0.5=-1/2`, the series `1-0.5+0.25-.......` is a geometric series, whose first term `a=1` and common ratio `r=-1/2` and `|r|<1`.

In a geometric series where first term is `a` and common ratio is `|r}<1`, sum up to `n^(th)` term `S_n` is given by

`S_n=(a(1-r^n))/(1-r)` and as in such infinite series as `n->oo`, the sum tends to `S=a/(1-r)`

Hence, here sum of the infinite geometric series `1-0.5+0.25-.......` is

`S=1/(1-(-1/2))=1/(1+1/2)=1/(3/2)=2/3`