Question

If `f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)...oo` then `f(1/2)=`?

Answer 1

The answer is `=2`

Explanation:

The series is

`f(x)=Pi_(k=0)^oo(1+x^(2^k))`

Let's start with `2` terms

`(1+x)(1+x^2)=1+x+x^2+x^3`

And with `3` terms

`(1+x)(1+x^2)(1+x^4)=1+x+x^2+x^3+x^4+x^5+x^6+x^7`

Therefore,

`f(x)=Pi_(k=0)^oo(1+x^(2^k))=1+x+x^2+x^3+x^4+x^5+x^6+.......................x^(oo)`

This is a geometric series

And

`f(x)=1/(1-x)` iif `|x|<1`

Therefore,

`f(1/2)=1/(1-1/2)=2`