Question

Given: $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90}, $ $ \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \cdots = \alpha, $ $ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \cdots = \beta $ Find the value of $ \frac{\alpha}{\beta} $.

• A. 15
• B. 14
• C. 23
• D. 18

Hint:

In series problems, especially when dealing with terms of even and odd powers, use the properties of symmetry in series to simplify the evaluation.

Answer 1

The Correct Option is A

- We are given that the series \(\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90}\). - \(\alpha\) is the sum of the series for odd powers: \[ \alpha = \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \cdots \] - \(\beta\) is the sum of the series for even powers: \[ \beta = \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \cdots \] We can express the sum of all terms as the sum of odd and even terms.
The sum for odd terms is half of the sum for all terms, and similarly for even terms.
Dividing these gives the ratio \( \frac{\alpha}{\beta} \), which results in 15.