If 1/1^4 + 1/2^4 + 1/3^4 + ... = pi^4/90, 1/1^4 + 1/3^4 + 1/5^4 + ... = a, 1/2^4 + 1/4^4 + 1/6^4 + ... = b, then a/b is equal to

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q6 – JEE-MAIN 2025

← Back

Home Maths Sequences and SeriesQuestion #415

\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots \infty = \frac{π^{4}}{90}

\frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + \dots \infty = α

\frac{1}{2^{4}} + \frac{1}{4^{4}} + \frac{1}{6^{4}} + \dots \infty = β

, then

\frac{α}{β}

is equal to

Clue 1 of 3: Series Decomposition

Recognize that the sum of all terms can be split into the sum of terms with odd indices and the sum of terms with even indices.

🧠 Full Solution Path

Step 1: Relate the given series✦ Active

Let the total sum be $S = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots \infty = \frac{π^{4}}{90}$ . The sum of terms with odd denominators is $α = \frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + \dots \infty$ . The sum of terms with even denominators is $β = \frac{1}{2^{4}} + \frac{1}{4^{4}} + \frac{1}{6^{4}} + \dots \infty$ . From these definitions, we can establish the relationship $S = α + β$ .

Step 2: Express

β

in terms of

S

○ Expand

The series $β$ consists of terms with even denominators. We can factor out a common term from each element:

β = \frac{1}{2^{4}} + \frac{1}{4^{4}} + \frac{1}{6^{4}} + \dots

β = \frac{1}{(2 \cdot 1)^{4}} + \frac{1}{(2 \cdot 2)^{4}} + \frac{1}{(2 \cdot 3)^{4}} + \dots

β = \frac{1}{2^{4}} (\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots)

β = \frac{1}{16} S

💡 Teacher's Secret Hint

Remember that $2^{4} = 16$ .

Step 3: Calculate

α

and the ratio

\frac{α}{β}

○ Expand

Using the relationship $S = α + β$ and the expression for $β$ from Step 2, we can find $α$ :

α = S - β = S - \frac{1}{16} S = (1 - \frac{1}{16}) S = \frac{15}{16} S

Now, we can calculate the required ratio $\frac{α}{β}$ :

\frac{α}{β} = \frac{\frac{15}{16} S}{\frac{1}{16} S} = 15

💡 Teacher's Secret Hint

The value of $\frac{π^{4}}{90}$ is not needed for the final ratio, as $S$ cancels out.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 6 – JEE-MAIN 2025

Clue 1 of 3: Series Decomposition

🧠 Full Solution Path