Express $\frac{1}{r(r+1)}$ in partial fractions.
Understand the Problem
The question asks to express the given fraction as partial fractions. This requires decomposing the fraction into simpler fractions. We will first identify the factors in the denominator and then express the fraction as a sum of simpler fractions over these factors.
Answer
$\frac{1}{r} - \frac{1}{r+1}$
Answer for screen readers
$\frac{1}{r} - \frac{1}{r+1}$
Steps to Solve
- Set up the partial fraction decomposition
We want to express $\frac{1}{r(r+1)}$ in the form $\frac{A}{r} + \frac{B}{r+1}$ where A and B are constants.
$$ \frac{1}{r(r+1)} = \frac{A}{r} + \frac{B}{r+1} $$
- Clear the denominators
Multiply both sides of the equation by $r(r+1)$ to eliminate the denominators:
$$ 1 = A(r+1) + Br $$
- Solve for A and B
We can solve for A and B by substituting suitable values for $r$.
Let $r = 0$:
$$ 1 = A(0+1) + B(0) \implies 1 = A $$ So, $A = 1$.
Let $r = -1$:
$$ 1 = A(-1+1) + B(-1) \implies 1 = -B $$ So, $B = -1$.
- Write the partial fraction decomposition
Substitute the values of A and B back into the partial fraction decomposition:
$$ \frac{1}{r(r+1)} = \frac{1}{r} + \frac{-1}{r+1} = \frac{1}{r} - \frac{1}{r+1} $$
$\frac{1}{r} - \frac{1}{r+1}$
More Information
Partial fraction decomposition is useful in calculus for integrating rational functions, and in other areas of mathematics for simplifying complex expressions.
Tips
A common mistake is to incorrectly set up the form of the partial fraction decomposition. For example, if the denominator had a repeated factor like $r^2$, the decomposition would need to include a term with $r^2$ in the denominator, as well as a term with r. Another common mistake is errors in solving for the constants A and B, especially when dealing with more complex equations.
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