**On Convergence of Infinite Sums**

Matthew Hanna

02/18/19

*“What is so special about infinite sums coming to a specific value?”*

We have all seen the classic sum.

Which is often represented as…

We, in our high school careers, have been taught that this sum is 2. This is

most often done so with an algebraic manipulation of both sides of the sum. We

haven’t been taught how to verify this sum. But how would we go about doing

this?

The basic idea is that if I end my sum at nite point, my result will deviate

from the actual sum by some error (which we will call ε. This error is some

positive real number. To verify that this sum indeed approaches 2, it suffices

to show that for any error, there exists a nite point in the series in which the

difference between 2 and this point is less than the error AND all further points

will also be less than the error.