Question

Find the sum of the series `1/2-1/3+1/4-1/9+1/8-1/27+........` to infinity?

Answer 1

`1/2-1/3+1/4-1/9+1/8-1/27+........=1/2`

Explanation:

`1/2-1/3+1/4-1/9+1/8-1/27+........` can be rewritten as

`(1/2+1/4+1/8+........)-(1/3+1/9+1/27+........)`

Hence, the given infinitive is one infinitive geometric series subtracted from another infinitive geometric series.

Observe that in the infinite series `1/2+1/4+1/8+........`, while first term `1/2` and common ratio is `1/2`

and in the infinite series `1/3+1/9+1/27+........`, first term `1/3` and common ratio is `1/3`

As for an infinite series whose first term is `a` and common ratio `r` is such that `|r|<1`, the sum converges to `a/(1-r)`.

Hence `1/2+1/4+1/8+........=(1/2)/(1-1/2)=(1/2)/(1/2)=1`, and

`1/3+1/9+1/27+........=(1/3)/(1-1/3)=(1/3)/(2/3)=1/2`

Hence `1/2-1/3+1/4-1/9+1/8-1/27+........=1-1/2=1/2`