The value of the sum 1/3² + 1/4² + 1/5² + ... ∞ is equal to

The sum of the series \( \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \ldots \) can be calculated using the formula for the Riemann zeta function, which is defined as \( \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \). For \( s = 2 \), the value of \( \zeta(2) \) is known to be \( \frac{\pi^2}{6} \).

Calculating the Series

To find the specific sum starting from \( n = 3 \), we can express it as:

• First, calculate \( \zeta(2) \): \( \zeta(2) = \frac{\pi^2}{6} \).
• Next, subtract the first two terms from \( \zeta(2) \): \( \frac{1}{1^2} + \frac{1}{2^2} = 1 + \frac{1}{4} = \frac{5}{4} \).

Final Calculation

Now, we can find the desired sum:

Sum = \( \zeta(2) - \left( \frac{1}{1^2} + \frac{1}{2^2} \right) = \frac{\pi^2}{6} - \frac{5}{4} \).

To simplify this, convert \( \frac{5}{4} \) to a fraction with a denominator of 24:

Sum = \( \frac{\pi^2}{6} - \frac{30}{24} = \frac{\pi^2 - 30}{24} \).

Thus, the value of the sum \( \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \ldots \) converges to \( \frac{\pi^2 - 30}{24} \).