Power Series x=0: What Does a0 & x0 Mean?

[kahwawashay1]

what does it mean if y(x)=$\sum$$^{∞}_{n=0}$ a_n(x-x₀)ⁿ
and y(0)=1 ?
does this mean that x₀=0 and a₀=1 ?

 

[lurflurf]

It means y(0)=1.

That x₀=0 and a₀=1

is one (obvious) possibility, but sometimes other choices are used.

For example we might chose a different x₀ is there is a singularity near zero and we need the series to converge is a certain range. Like if there were a singularity at x=-1 and we would like a series valid on (-1,3) we might choose x₀=2. Then y(0)=1 means

1=y(0)=$\sum$$^{∞}_{n=0}$ a_n(-2)ⁿ

A bit tricky to work with, but not too bad.