Question

How do you find the sum of the convergent series 0.9-0.09+0.009-...? If the convergent series is not convergent, how do you know?

Answer 1

Note that the common ratio has an absolute value less than one, and use the geometric series formula to find that

`0.9-0.09+0.009-... = 9/11 = 0.bar(81)`

Explanation:

A geometric series is a series of the form

`a + ar + ar^2 + ar^3 + ... = sum_(n=0)^ooar^n`

where `a` is the initial term of the series and `r` is the common ratio between terms.

If `|r| < 1`, then the sum is given by

`sum_(n=0)^ooar^n = a/(1-r)`

An explanation of where this sum comes from is given in this answer.

For the given series, we have

`0.9 -0.09 +0.009 -... = 9/10 + 9/10(-1/10) + 9/10(-1/10)^2 + ...`

`= sum_(n=0)^oo9/10(-1/10)^n`

Thus `a = 9/10` and `r = -1/10`.
As `|r| = 1/10 < 1` we have the sum

`0.9 -0.09 +0.009 -...= sum_(n=0)^oo9/10(-1/10)^n`

`= (9/10)/(1-(-1/10))`

`=9/11 = 0.bar(81)`