Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{j=3}^{5} \frac{1}{j^{2}-3}$$

Step-by-Step Solution

Answer

The sum of the values calculated based on the values of j from 3 to 5 is approximately 0.259.

Step 1: Identify the Range of j

In the given summation, j ranges from 3 to 5. For these values of j, substitute them into the given formula, \( \frac{1}{j^{2}-3} \) and compute the sum.

Step 2: Substitute j=3

When j=3, the formula becomes \( \frac{1}{3^{2}-3} \) which simplifies to \( \frac{1}{6} \).

Step 3: Substitute j=4

When j=4, the formula becomes \( \frac{1}{4^{2}-3} \) which simplifies to \( \frac{1}{13} \).

Step 4: Substitute j=5

When j=5, the formula becomes \( \frac{1}{5^{2}-3} \) which simplifies to \( \frac{1}{22} \).

Step 5: Calculate the Sum

Now sum up these fractions: \( \frac{1}{6} + \frac{1}{13} + \frac{1}{22} \). The sum comes out to be approximately 0.259.