Evaluate the Summation sum from j=3 to 5 of 1/(j^2-3)
The problem provided presents a mathematical expression that requires evaluation. Specifically, it is asking for the calculation of a definite summation (also known as a finite sum). The summation operation is to be performed from j=3 to j=5. For each integer value of j within this range, the expression to sum is 1/(j^2-3). This involves taking the reciprocal of the quantity obtained by squaring the integer j and subtracting 3 from that result. The overall task is to calculate the sum of these reciprocals for each value of j between 3 and 5, inclusive.
$\sum_{j = 3}^{5} \frac{1}{j^{2} - 3}$
Answer
Solution:
Step 1: Write out the summation for each term of j.
$\frac{1}{3^2 - 3} + \frac{1}{4^2 - 3} + \frac{1}{5^2 - 3}$
Step 2: Begin simplification.
Step 2.1: Identify a common denominator.
Step 2.1.1: Multiply $\frac{1}{3^2 - 3}$ by $\frac{(4^2 - 3)(5^2 - 3)}{(4^2 - 3)(5^2 - 3)}$.
$\frac{(4^2 - 3)(5^2 - 3)}{(3^2 - 3)(4^2 - 3)(5^2 - 3)} + \frac{1}{4^2 - 3} + \frac{1}{5^2 - 3}$
Step 2.1.2: Multiply $\frac{1}{4^2 - 3}$ by $\frac{(3^2 - 3)(5^2 - 3)}{(3^2 - 3)(5^2 - 3)}$.
$\frac{(4^2 - 3)(5^2 - 3)}{(3^2 - 3)(4^2 - 3)(5^2 - 3)} + \frac{(3^2 - 3)(5^2 - 3)}{(4^2 - 3)(3^2 - 3)(5^2 - 3)} + \frac{1}{5^2 - 3}$
Step 2.1.3: Multiply $\frac{1}{5^2 - 3}$ by $\frac{(3^2 - 3)(4^2 - 3)}{(3^2 - 3)(4^2 - 3)}$.
$\frac{(4^2 - 3)(5^2 - 3)}{(3^2 - 3)(4^2 - 3)(5^2 - 3)} + \frac{(3^2 - 3)(5^2 - 3)}{(4^2 - 3)(3^2 - 3)(5^2 - 3)} + \frac{(3^2 - 3)(4^2 - 3)}{(5^2 - 3)(3^2 - 3)(4^2 - 3)}$
Step 2.1.4: Simplify the denominators to a common denominator.
$\frac{(4^2 - 3)(5^2 - 3) + (3^2 - 3)(5^2 - 3) + (3^2 - 3)(4^2 - 3)}{(3^2 - 3)(4^2 - 3)(5^2 - 3)}$
Step 2.2: Simplify the numerator.
$\frac{13(5^2 - 3) + 6(5^2 - 3) + 6(4^2 - 3)}{(3^2 - 3)(4^2 - 3)(5^2 - 3)}$
Step 2.3: Perform the arithmetic operations.
$\frac{286 + 132 + 78}{(3^2 - 3)(4^2 - 3)(5^2 - 3)}$
Step 2.4: Combine like terms.
$\frac{496}{(3^2 - 3)(4^2 - 3)(5^2 - 3)}$
Step 2.5: Simplify the denominator.
$\frac{496}{6 \cdot 13 \cdot 22}$
Step 2.6: Simplify the fraction.
$\frac{496}{1716}$
Step 2.7: Reduce the fraction to lowest terms.
$\frac{124}{429}$
Step 3: Present the result.
Exact Form: $\frac{124}{429}$ Decimal Form: Approximately $0.289$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a basic concept in calculus and algebra. The summation notation $\sum$ represents the addition of terms in a sequence, with the lower and upper bounds indicating the start and end of the summation, respectively.
To solve the problem, we expand the series for each value of $j$ within the given range. We then aim to combine these fractions by finding a common denominator. This process involves algebraic manipulation, including multiplying fractions to achieve a common denominator.
The arithmetic operations performed include raising numbers to powers (exponentiation), multiplication, and subtraction. After combining the terms over a common denominator, we simplify the numerator and denominator separately.
Finally, we reduce the fraction to its lowest terms by canceling out common factors. This involves factoring out common divisors and simplifying the fraction to its simplest form.
Understanding how to manipulate algebraic expressions, perform arithmetic operations, and simplify fractions are key skills utilized in this problem-solving process.
