# How do you use summation notation to expression the sum 15-3+3/5-...-3/625?

Mar 7, 2017

#### Explanation:

Let try to fill in the finite sum:

$15 - 3 + \frac{3}{5} - \frac{3}{25} + \frac{3}{125} - \frac{3}{625}$

If your remove a common factor of 15, you get:

$1 - \frac{1}{5} ^ 1 + \frac{1}{5} ^ 2 - \frac{1}{5} ^ 3 + \frac{1}{5} ^ 4 - \frac{1}{5} ^ 5$

You can write 1 as $\frac{1}{5} ^ 0$:

$\frac{1}{5} ^ 0 - \frac{1}{5} ^ 1 + \frac{1}{5} ^ 2 - \frac{1}{5} ^ 3 + \frac{1}{5} ^ 4 - \frac{1}{5} ^ 5$

Because the minus sign appears on the odd powers, one can see that we are raising -5 to a negative power:

$\frac{1}{5} ^ 0 - \frac{1}{5} ^ 1 + \frac{1}{5} ^ 2 - \frac{1}{5} ^ 3 + \frac{1}{5} ^ 4 - \frac{1}{5} ^ 5 = {\sum}_{n = 0}^{5} - {5}^{-} n$

To obtain the original sum, multiply by 15:

$15 - 3 + \frac{3}{5} - \frac{3}{25} + \frac{3}{125} - \frac{3}{625} = 15 {\sum}_{n = 0}^{5} - {5}^{-} n$