Question

How can an infinite series have a finite sum?

Answer 1

Hint: To answer this question, we need to take a real life example. We will then convert this real life example in the form of geometric series. As we know that the infinite geometric series has a certain sum, we can prove that an infinite series has a finite sum.

Complete step by step solution:
Let us consider the runner who has to run the distance of $ 2km $ .
Now, the condition is that the runner has to take a stop at half of the remaining distance from the finish line.
As he starts, his first stop will be at $ 1km $ . Now, he has one more kilometer to cover. Thus, his second stop will be at $ \dfrac{1}{2}km $ . Similarly, the third stop will be at \[\dfrac{1}{4}km\] and this goes on if the runner keeps stopping in this way. However, we know that the total distance to be covered by him is $ 2km $ .
This total distance can be expressed as the geometric series:
 $ 1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + ...... = {\sum\limits_{n = 0}^\infty {\left( {\dfrac{1}{2}} \right)} ^n} $
Since the runner is taking a stop at each half of your remaining distance, he will never actually reach the finish line, but he is getting closer and closer to it. Now, it is clear that the total distance is $ 2km $ , therefore, the sum of this geometric series will be 2.
Thus, we can say that an infinite series can have a finite sum.


Note: We know that for convergent geometric series, its sum can be found by using the formula:
 $ S = \dfrac{a}{{1 - r}} $ , where, $ a $ is the first term and $ r $ is the common ratio of the geometric series.
In our case, $ a = 1 $ and $ r = \dfrac{1}{2} $ .
 $ \Rightarrow S = \dfrac{1}{{1 - \dfrac{1}{2}}} = \dfrac{1}{{\dfrac{1}{2}}} = 2 $
This also indicates that the infinite series can have a finite sum.