If 1/1⁴ + 1/2⁴ + 1/3⁴ + … ∞= π⁴/90 ,
1/1⁴ + 1/3⁴ + 1/5⁴ + … ∞= α ,
1/2⁴ + 1/4⁴ + 1/6⁴ + … ∞= β ,
then α/β is equal to :

Solution

Maths sir · 4 steps

Every even term is just an ordinary term divided by 16, so β is one-sixteenth of the full sum.

1. Name the full sum. Let S = 1/1⁴ + 1/2⁴ + 1/3⁴ + … = π⁴/90. Splitting the terms into odd and even denominators gives S = α + β.
2. Pull out the 2. Each even term is 1/(2n)⁴ = 1/16 · 1/n⁴. So β is the whole series S scaled down by 16.

   β = 1/16(1/1⁴ + 1/2⁴ + 1/3⁴ + …) = S/16

3. Find alpha. α = S - β = S - S/16 = 15S/16.
4. So. α/β = (15S/16)/(S/16) = 15. The value π⁴/90 is never needed, and the answer is (d).

The idea to keep. For Σ 1/nᵏ, the even terms add up to 1/2ᵏ of the full sum, and the odd terms make up the rest. So odd : even is always (2ᵏ - 1) : 1.

Answer from the NTA final answer key. Solution by PrepWiser.