document.write( "Question 1074919: show that the following series converges absolutely for |x|<1 and compute the sum.
\n" ); document.write( "f(x)=1-x-x^2+x^3-x^4-x^5+x^6-x^7-x^8+...
\n" ); document.write( "The hint we were given was to write f(x) as the sum of 3 geometric series with a common ratio of x^3 \n" ); document.write( "

Algebra.Com's Answer #689754 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "is one of the infinite geometric series hinted at.
\n" ); document.write( "The first term is $\"1\"$ and the common ratio is $\"x%5E3\"$ .
\n" ); document.write( "The other two series are
\n" ); document.write( "

,
\n" ); document.write( "with first term $\"-x\"$ , and
\n" ); document.write( "

,
\n" ); document.write( "with first term $\"-x%5E2\"$ .
\n" ); document.write( "Geometric series with a first term $\"b\"$ ,
\n" ); document.write( "and common ratio $\"r\"$ such that $\"abs%28r%29%3C1\"$
\n" ); document.write( "converge to $\"b%2F%281-r%29\"$ .
\n" ); document.write( "Since $\"abs%28x%29%3C1\"$ means $\"abs%28x%5E3%29%3Cabs%28x%29%3C1\"$ ,
\n" ); document.write( "those 3 series converge to the sums shown above.
\n" ); document.write( "Since each series converges, so does its sum.
\n" ); document.write( " $\"f%28x%29=S%5B1%5D%2BS%5B2%5D-S%5B3%5D=%281-x-x%5E2%29%2F%281-x%5E3%29\"$ . \n" ); document.write( "

\n" );