Calculus Examples

$\sum_{n = 1}^{\infty} \frac{1}{4 n^{2} - 1}$

Step 1

Simplify the denominator.

Tap for more steps...

Step 1.1

Rewrite $4 n^{2}$ as ${(2 n)}^{2}$ .

$\frac{1}{{(2 n)}^{2} - 1}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{1}{{(2 n)}^{2} - 1^{2}}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 n$ and $b = 1$ .

$\frac{1}{(2 n + 1) (2 n - 1)}$

Step 2

Rewrite the summation.

$\sum_{n = 1}^{\infty} \frac{1}{(2 n + 1) (2 n - 1)}$