Imaginary Common Ratio?
Algebra
Level
1
True or false:
\[ 1 + i + i^2 + i^3 + i^4 + \cdots \]
The expression above represents an infinite geometric progression sum with first term, \(a = 1\) and common ratio \(r = i \). And it can be expressed as \[ \dfrac a{1-r} =\dfrac 1{1-i} = \dfrac 1{1-i} \cdot \dfrac{1+i}{1+i} = \dfrac{1+i}{1-i^2} = \dfrac12 + \dfrac12 i \; .\]
Clarification: \(i=\sqrt{-1}\).
by
Chung Kevin