SOLUTION: show that the following series converges absolutely for |x|<1 and compute the sum. f(x)=1-x-x^2+x^3-x^4-x^5+x^6-x^7-x^8+... The hint we were given was to write f(x) as the sum of

Question 1074919: show that the following series converges absolutely for |x|<1 and compute the sum.
f(x)=1-x-x^2+x^3-x^4-x^5+x^6-x^7-x^8+...
The hint we were given was to write f(x) as the sum of 3 geometric series with a common ratio of x^3
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

is one of the infinite geometric series hinted at.
The first term is and the common ratio is .
The other two series are
,
with first term , and
,
with first term .
Geometric series with a first term ,
and common ratio such that
converge to .
Since means ,
those 3 series converge to the sums shown above.
Since each series converges, so does its sum.
.
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