Question

How can an infinite series have a finite sum?

Answer 1

Yes, the same question used to bother me as well, but the following example might make you feel more comfortable with it.

Mark the start line and the finish line 2 m apart, then take each step 1/2 of your remaining distance from the finish line; for example, your first step is 1 m, your second step is 1/2 m, your third step is 1/4 m, and so on. If you keep going forever, the total distance you will travel can be expressed as the geometric series below.
`1+1/2+1/4+...=sum_{n=0}^{infty}(1/2)^n`
Since you are taking 1/2 of your remaining distance, you will never actually reach the finish line, but you are getting closer and closer to it. Now, it is clear that the total distance approaches 2 m; therefore, the sum of the geometric series above is 2.

If you remember that the sum of a convergent geometric series can be found by `a/{1-r}`, then you can verify
`sum_{n=0}^{infty}(1/2)^n=1/{1-1/2}=2`,
which agrees with our previous intuitive argument.

Are you now convinced that an infinite series can have a finite sum?