Converging product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?
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Problem



It is well known that if $| x | < 1$ then 1+x+x2++xn=1xn+11x and hence (taking limits) we have the sum of the infinite geometric series 1+x+x2++xn+=11x We are now going to obtain a similar formula for an infinite product, namely (1+x)(1+x2)(1+x4)(1+x8)(1+x2n)=11x Evaluate the product (1x)(1+x)(1+x2)(1+x4)(1+x8) Show, by induction, that (1+x)(1+x2)(1+x2n)=1x2n+11x and hence (taking limits) the given formula for the infinite product follows.