If frac 1 1 4 +frac 1 2 4 +frac 1 3 4 +ldots ∞=frac π 4 90 , frac 1 1 4 +frac 1 3 4 +frac 1 5 4 +ldots ∞=α frac 1 2 4 +frac 1 4 4 +frac 1 6 4 +ldots ∞=β , Then α/β is equal to

Mathematics · Sequence and Series

JEE Main 2025 — 8 April, Evening Shift — Question 31

If $\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\ldots \infty=\frac{\pi^{4}}{90}$ , $\frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\ldots \infty=\alpha$ $\frac{1}{2^{4}}+\frac{1}{4^{4}}+\frac{1}{6^{4}}+\ldots$

$\infty=\beta$ , Then $\frac{\alpha}{\beta}$ is equal to

Option A:
14
Option B:
15
Correct
Option C:
18
Option D:
23

Answer: B

Step-by-step solution

$\alpha=\frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\ldots$

\begin{aligned} & \beta=\frac{1}{2^{4}}+\frac{1}{4^{4}}+\frac{1}{6^{4}}+\ldots \\& =\frac{1}{2^{4}}\left[\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+\ldots\right] \end{aligned}

$\Rightarrow 16 \beta=\left[\frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\ldots\right]+\left[\frac{1}{2^{4}}+\frac{1}{4^{4}}+\frac{1}{6^{4}}+\ldots\right]$ $=\alpha+\beta$

$\Rightarrow 15 \beta=\alpha \Rightarrow \frac{\alpha}{\beta}=15$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 8 April, Evening Shift
Subject: Mathematics
Chapter: Sequence and Series
Topic: Introduction to Sequence and Series