# Evaluate the infinite series

1. Oct 15, 2008

### kingwinner

1. The problem statement, all variables and given/known data
Evaluate

∑ [(e-15 15x) / x!]
x=16

15
∑ [(e-15 15x) / x!]
x=0

2. Relevant equations

3. The attempt at a solution
The only way I can think of is writing out every term explicitly and adding them on a calculator.
Is there any faster way (without having to write out every term explicitly) to calculate the above sums?

Thanks for any help!

2. Oct 15, 2008

### Dick

The sum of the two series is e^(-15)*e^(15), right? If you want the sums individually you do need a calculator. If you want the total, it's pretty obvious.

3. Oct 15, 2008

### Staff: Mentor

∑ [(e-15 15x) / x!]
x=0

= e-15 *

∑ (15x) / x!
x=0

= 1, if that's any help.

4. Oct 16, 2008

### HallsofIvy

Once you know the sum of the two series, since the first is finite, it's not all that hard to find the sum of 15x/x! for x from 0 to 15 by hand and then get the other sum by subtracting. The only place you really NEED a calculator (though I would recommend it for the tedious multiplications, divisions, and subtractions) is to evaluate e-15

5. Oct 16, 2008

$$\sum_{x=16}^\infty \left(\frac{e^{-15} 15^x}{x!}\right)$$
is $$\Pr(X \ge 16)$$, the other sum is $$\Pr(X \le 15)$$.