## Example 8.4.5: A Taylor Series that does not Converge to its Function

Define
. Show that:

- The function is infinitely often differentiable
- The Taylor series
Taround_{g}(x, 0)c = 0has radius of convergence infinity.- The Taylor series
Taround_{g}(x, 0)c = 0does not converge to the original function.

We have already shown before that:

*g*is infinitely often differentiable*g*for all^{(n)}(0) = 0*n*

The Taylor series of *g* is therefore:

T_{g}(x, 0) = a_{n}x^{n}= 1/n! g^{(n)}(0) x^{n}= 0 x^{n}= 0 for all x

In particular, the radius of convergence is infinity. But the original
function is not identically equal to zero (*g(0) = 0* but
*g(x) > 0* for *x # 0*), so that:

T_{g}(x, 0) # g(x) unless x = 0