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?

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Answer 1 · 2015-12-30T00:29:29

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)#

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)#