Using the integral test, how do you show whether #sum (1/e^k)# diverges or converges?
1 Answer
The series converges as proved by the integral test explained below.
Explanation:
The integral test states that:
If
First we have to look at the nature of
graph{1/e^x [-10, 10, -5, 5]}
As we can see
Remember
Integrate this with respect to
For the upper limit, we can see that as
For the lower limit we simply obtain:
So evaluating the limits gives:
So, by the integral test, as the integral converges to a finite value then the summation:
It is important to note that the integral cannot be used to evaluate the sum , but only test whether it converges or not, that is:
Infact if we evaluate the sum we get: