Solving the Paradox of 0^0 in Power Series Analysis

[quasar987]

If I evaluate the power serie

y(x) = \sum_{n=0}^{\infty}a_nx^n

at x = 0, the first term is a_0(0)^0. But 0^0 is undefined, is it not? How is this paradox solved?

 

[Hurkyl]

Because that notation for power series doesn't really mean you're supposed to compute 0^0 to get the n=0 term.

Though, some sources will prefer to write that series as

y(x) = a_0 + \sum_{n=1}^{\infty}a_nx^n

to remove any confusion. (The two mean the same thing, though)

 

[quasar987]

I didn't know that. They don't tell us about any of the subtelties of anything at school. They're totally useless.