Pacman! Eat Up This Infinite Product
Algebra
Level
3
Simplify the infinite product P(x) = (1+a)(1+a2)(1+a4)(1+a8)(1+a16)..., given |a| < 1. Post your answer as P(1/2) using the simplified product. Express as a decimal if necessary.
The answer is 2.
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1 solution
Multiply both sides by (1-a) and bracket it with (1+a) on the right side.
(1-a)P(a) = {(1-a)(1+a)} x (1+a^2)(1+a^4)(1+a^8)(1+a^16)...
The two terms in brackets combine to form (1-a^2). Substitute this for (1-a)(1+a) and bracket it with (1+a^2).
(1-a)P(a) = {(1-a^2)(1+a^2)} x (1+a^4)(1+a^8)(1+a^16)...
The two terms in brackets combine to form (1-a^4). Substitute this for (1-a^2)(1+a^2) and bracket it with (1+a^4).
(1-a)P(a) = {(1-a^4)(1+a^4)} x (1+a^8)(1+a^16)...
Each term in the infinite product “eats” the next one, with the end result that:
(1-a)P(x) = 1 or
P(x) = 1/(1-a)
P(1/2) = 1/(1-½) = 2