Recent content by V150

Oh, oh. I finally got it. f'(x) is zero because f(x) = 0 and f'(x) is like differentiating zero, right?

So the question would be, "why c_{n} is 0 for all n when \sum_{n=0}^{\infty}c_{n}x^{n}=0 for all x with the radius of convergence R>0?

Yes, for all x with the radius of convergence R>0.

hmmm. How can I use that to prove \sum_{n=0}^{\infty}c_{n}x^{n}=0,then c_{n}=0 when x is all real numbers?

@AlephZero What I want to know is why c_n = 0 have to be zero if \sum_{n=0}^{\infty}c_{n}x^{n} = 0 , not 'IF' c_n=0 . Would you make this more concrete, please?

Homework Statement Suppose that f(x)=\sum_{n=0}^{\infty}c_{n}x^{n}for all x. If \sum_{n=0}^{\infty}c_{n}x^{n} = 0, show that c_{n} = 0 for all n. Homework Equations The Attempt at a Solution I know, by using taylor expansion, c_{n}=\frac{f^{n}(0)}{n!}, and because...