How can an infinite series have a finite sum?

Question

Isaiah Allen · Accepted Answer

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?

Samantha Adkison · Answer

undefined An infinite series can have a finite sum if the terms of the series approach zero sufficiently quickly as the series progresses. This concept is based on the idea of convergence in mathematics.

Mathematically, an infinite series $\sum_{n=1}^{\infty} a_n$ converges to a finite sum $S$ if the sequence of partial sums $S_N = \sum_{n=1}^{N} a_n$ approaches $S$ as $N$ approaches infinity.

Several tests are used to determine the convergence of infinite series, such as the ratio test, root test, and comparison test. If these tests confirm that the series converges, then its sum is finite.

For example, the geometric series $ \sum_{n=0}^{\infty} r^n $ converges to $ \frac{1}{1-r} $ if $ |r| < 1 $, even though it has infinitely many terms. This is because as more terms are added, each term becomes smaller, and the sum approaches a finite value.