And for an infinite series where | r | < 1 (absolute value of r is less than one), you can get a finite value. But how can this be? Let’s look back at this equation.
Sn = a(1 - r^n) / (1 - r)
When n tends towards infinity, it becomes very big. And since r is very small, r^n tends towards zero. You can try it out for yourself, typing a positive number less than 1 to the power of a really big number nets you a very very small number. As n becomes closer to infinity, r^n becomes closer to 0. So we can substitute r^n with zero like this:
Sn = a(1 - 0) / (1 - r)
= a / (1 - r)
And since this both a, the first term, and r, the common ratio, is finite, Sn must also be finite! And to go back to Zeno’s paradox. Let’s say a = 1 and r = 1/2. This means:
Sn = 1 / (1 - r)
= 1 / (1 - 0.5)
= 1 / 0.5
= 2
You find that Sn is the finite value 2. Maths is cool!
This, however, does not work if the absolute value of r is greater than or equal to one. The sum of all terms for such a geometric sequence would not be finite. (think 2 + 4 + 8 + …, the total sum is infinite as r = 2)
And for an infinite series where | r | < 1 (absolute value of r is less than one), you can get a finite value. But how can this be? Let’s look back at this equation.
Sn = a(1 - r^n) / (1 - r)
When n tends towards infinity, it becomes very big. And since r is very small, r^n tends towards zero. You can try it out for yourself, typing a positive number less than 1 to the power of a really big number nets you a very very small number. As n becomes closer to infinity, r^n becomes closer to 0. So we can substitute r^n with zero like this:
Sn = a(1 - 0) / (1 - r)
= a / (1 - r)
And since this both a, the first term, and r, the common ratio, is finite, Sn must also be finite! And to go back to Zeno’s paradox. Let’s say a = 1 and r = 1/2. This means:
Sn = 1 / (1 - r)
= 1 / (1 - 0.5)
= 1 / 0.5
= 2
You find that Sn is the finite value 2. Maths is cool!
This, however, does not work if the absolute value of r is greater than or equal to one. The sum of all terms for such a geometric sequence would not be finite. (think 2 + 4 + 8 + …, the total sum is infinite as r = 2)