# Prove that the sum of the infinite series #(1*3)/2 + (3*5)/(2^2) + (5*7)/(2^3) + (7*9)/(2^4) ........ oo = 23#?

##### 2 Answers

The general term is

#sum_(n = 1)^oo ((2n - 1)(2n + 1))/2^n#

Which can be rewritten as

#sum_(n = 1)^oo (4n^2 - 1)/2^n#

Which in turn can be written as

#sum_(n = 1)^oo (4n^2)/2^n - 1/2^n#

#sum_(n = 1)^oo 2^2/2^n n^2 - 1/2^n#

#sum_(n = 1)^oo 2^(2 - n) n^2 - 1/2^n#

We know this first sequence will converge. This is because

According to wolfram alpha, the sum is

The second series is just your run of the mill geometric series.

#s_oo = (1/2)/(1 - 1/2)#

#s_oo = (1/2)(2)#

#s_oo = 1#

So the sum of the entire sequence will be

Hopefully this helps!

See below.

#### Explanation:

Here

then, making