Trying to do a math proof in a comment is hard, but I’ll try my best:
A term in a geometric series is defined as: u =ar^n-1
where a is the first term and r is the common ratio (the multiplier you use to get the next term)
so the sequence is a, ar, ar^2, … , ar^(n-1) where n is the number of terms in the sequence.
The sum of all the terms in a sequence, the geometric series, can be found using this for (absolute) values of r that are below 1:
Sn = a(1 - r^n) / (1 - r)
for (absolute) values of r above 1, it looks like this:
Sn = a(r^n - 1) / (r - 1)
But both equations will work with any value of r, they are just rearranged to make the maths easier. Where n is the number of terms in the series. For example, the series 2, 4, 8, 16 has four terms, the common ratio of 2 and a first term of 2. This means:
Sn = 2(2^4 - 1) / (2 - 1)
= 2(16 - 1) / 1
= 2 * 15
= 30
And if you check 2 + 4 + 8 + 16, you will find that it correctly equals 30, meaning the maths is right! In this case, it would be easier to add them up individually, but using the formula is useful when you have a large series with many different terms or when you have limited information (i.e. you are not given every individual term)
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!
Trying to do a math proof in a comment is hard, but I’ll try my best:
A term in a geometric series is defined as: u =ar^n-1
where a is the first term and r is the common ratio (the multiplier you use to get the next term)
so the sequence is a, ar, ar^2, … , ar^(n-1) where n is the number of terms in the sequence.
The sum of all the terms in a sequence, the geometric series, can be found using this for (absolute) values of r that are below 1:
Sn = a(1 - r^n) / (1 - r)
for (absolute) values of r above 1, it looks like this:
Sn = a(r^n - 1) / (r - 1)
But both equations will work with any value of r, they are just rearranged to make the maths easier. Where n is the number of terms in the series. For example, the series 2, 4, 8, 16 has four terms, the common ratio of 2 and a first term of 2. This means:
Sn = 2(2^4 - 1) / (2 - 1)
= 2(16 - 1) / 1
= 2 * 15
= 30
And if you check 2 + 4 + 8 + 16, you will find that it correctly equals 30, meaning the maths is right! In this case, it would be easier to add them up individually, but using the formula is useful when you have a large series with many different terms or when you have limited information (i.e. you are not given every individual term)
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!