# Power series, x=0

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

lurflurf
Homework Helper
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}$ an(-2)n

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