An Interesting Geometric-like Series

Evaluating a simple, yet interesting infinite sum.

[@danielc.martin]

I think this is named as "step series" or "gabriel's series" and it is an special case of aritmo-geometric series.

[@vishtoxic5928]

This summation only holds truth if -1

[@joetsu7109]

Yes otherwise the summation would diverge

[@satho6440]

How are you able to pull x out of the summation as its a variable and not a constant? Also how do you know the summation is converging and not diverging?

[@Mathority1729]

We can pull an x out of the summation because x is independent of the summation, the variable dependent on the summation here is n. x just represents a ratio such as 1/2 or 1/3, etc. In fact, this series only converges if |x|

[@skyinhellandheaven]

I think you overlooked a negative sign because (1/u)' = -u'/u², or am I missing something?!

[@Mathority1729]

Yes but in this case since u=1-x, we’re finding the derivative of 1/(1-x) or (1-x)^(-1) And because of the chain rule, we also must multiply that result by the derivative of (1-x), which is -1. That cancels out the negative you mentioned Hopefully that makes sense now! Thank you so much for watching! Really appreciate it 😄

[@skyinhellandheaven]

@Mathority1729 I understand now, thanks! Great video btw.