Example: Since
1+r+r2+r3+…=11−r,
for |r|<1, replacing r with x gives us the first example of a function expressed as power series, namely
f(x)=11−x=∞∑n=0xn=1+x+x2+… as long as |x|<1.
Similarly, replacing r=−x gives us
11+x=∞∑n=0(−x)n=1−x+x2+… as long as |x|<1.
and replacing r=x2 yields
11+x2=∞∑n=0(−x2)n=1−x2−x4+x6+… as long as |x2|<1. That is, as long as |x|<1.
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