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?
+33615 Multiply S by 23 so 8S = (2/2)3 + (2/4)3 + (2/6)3 + ... = 1/13 + 1/23 + 1/33 + ... = S + T
So: 8S = S + T
Divide both sides by T and rearrange. I’ll leave you to do this.
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 \(S/T = \boxed{1/7}.\)
