Let S = \(\frac{1}{2^3}\) + \(\frac{1}{4^3}\) + \(\frac{1}{6^3}\) + \( \dotsb\) and T = \(\frac{1}{1^3}\) + \(\frac{1}{3^3}\) + \(\frac{1}{5^3}\) + \(\dotsb\). Find S / T. Hint(s): Consider the sum S + T. Can you relate this sum to S or T?

Guest Jun 24, 2019

edited by
Guest
Jun 24, 2019

#1**+4 **

Best Answer

Multiply S by 2^{3} so 8S = (2/2)^{3} + (2/4)^{3} + (2/6)^{3} + ... = 1/1^{3} + 1/2^{3} + 1/3^{3} + ... = S + T

So: 8S = S + T

Divide both sides by T and rearrange. I’ll leave you to do this.

Alan Jun 24, 2019

#3**0 **

We have that

\(S + T = \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \dotsb.\)

Dividing both sides by \(8 = 2^3,\) we get

\(\frac{S + T}{8} = \frac{1}{2^3} + \frac{1}{4^3} + \frac{1}{6^3} + \frac{1}{8^3} + \dots = S.\)

Rearranging this equation, we get \(S/T = \boxed{1/7}.\)

Guest Jun 24, 2019