geometric series help

[fickle anvil]

You can call the series Sn which give you.

Sn = 1 + a + a^2 + … a^n

You can then say

a*Sn = a + a^2 + a^3 + … a^(n + 1)

Subtract both you will get.

Sn - a*Sn = 1 - a^(n + 1)

(1 - a)Sn = 1 - a^(n + 1)

Sn = [ 1 - a^(n + 1) ] / [ 1 - a ]

Since |a| < 1, limit of Sn to infinity will get you.

Sn = 1 / ( 1 - a)

Because a^(n + 1) will become 0.

So the series converge if |a| < 1.

For |a| not equal to one, you can plug in the series and see it start to oscillate and grow instead of converge for a < -1 so therefore the series diverge.

a > 1 the series goes to inf

It better to show both negative and positive side and describe the different. If other people reading this doc then don’t use - because I confused it for negative sign.