Summing a Geometric Series: Can We Use the Formula 1/(1-x)?

[Mathman23]

Hi

Can I claim that in order to find the sum of the series:

\sum_{n = 0} ^{\infty} 2^{- n}

\sum_{n = 0} ^{\infty} 2^{- n} = \sum_{n = 0} ^{\infty} x^n = \frac{1}{1-x} ?


Sincerely Yours
Fred

 

[matt grime]

No, you can't claim *that* (since it is false; one side is a number, the other is a power series in x), but you can use the series if you do so legitimately.

 

[HallsofIvy]

In other words, since 2^-n= (2^-1)ⁿ, yes, if x= 2^-1. (Assuming, of course, that the sum converges. Can you show that?)