Suppose there is a real sequence $(c_i)_{i=0}^\infty$ such that $\sum_{i=0}^\infty c_ix^i$ converges everywhere on some interval (a,b). Let (s,t) be an interval such that $(s,t)\subset(a,b)$ and such that $\sum_{i=0}^\infty c_ix^i=0$ everywhere on (s,t). Does it follow that $\sum_{i=0}^\infty c_ix^i=0$ everywhere on (a,b) or, is it possible for the sum to be non-zero on (a,s) or (t,b)? I feel like this question ought to be trivial but I can't seem to work out the answer.
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2Yes, the function must be identically zero in the setting you describe. See here. – Andrés E. Caicedo Aug 02 '13 at 21:28
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This fails for the other common way of expanding functions, as trigonometric series: A function $f(\theta)=b_0+\sum_{n=1}^\infty(a_n\sin(n\theta)+b_n\cos(n\theta))$ may be zero for all $\theta$ in some proper subinterval of $[0,2\pi)$, without being zero everywhere. – Andrés E. Caicedo Aug 02 '13 at 21:32
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No,
An analytic function (function locally expressable as a power series) can be expanded as a power series everywhere in it's domain. This means you can expand it around a point in (s, t)$. But if you do this, since it is zero, then all derivaties are zero and thus the coefficients of the power series must be zero. In fact, more generally, if there are two power series f(x) and g(x) that agree on a set with a limit point then they agree everywhere.
Owen Sizemore
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