I know that, by long division, or binomial formula or Taylor formula that this function can be developed into the geometric series: $1/(1-x)=1+x+x^2+x^3+x^4+\ldots $ and I thought that this is the only series that represent the function mentioned above. However, recently, I read "Traite Elementaire des Series" by Eugene Catalan and he said, on page 60, that "The same function can admit multiple expansions". He gives the series below:
I am just curious how can we derive this second series. So a function may admit multiple series representation? This is something new that I don't know. Maybe a better way to think about this is that two or multiple series can converge to the same function, which is the same as multiple numerical series may converge to the same sum.