Show coefficients of two power series agree where they coincide on an interval Suppose two power series centered at zero have radii of convergence $4$, and the values of the two series coincide on the interval (3,4). Show that the coefficients of the two power series agree. Any hint will be appreciated.
I tired setting $x=0,$ but since we don't know if they agree at $0,$ it is tricky. I tried to shift the function, but it doesn't seem to help.
 A: This is an application of the fact, that if a power series is zero on some interval, then its coefficients are all zero and its in fact the zero function. (see here, or here, and here, maybe here)

Let $f(x) = \sum_{k=0}^\infty c_k x^k$ and $g(x) = \sum_{k=0}^\infty d_kx^k$ be two power series with $c_k,d_k \in \mathbb R$ and radius of convergence $R_f,R_g \gt 0$. If $f(x)=g(x)$ for all $|x| \lt R$, then $c_k=d_k$ for all $k$ and $R_f=R_g$.

Define
$$h(x) := f(x)-g(x) = \sum_{k=0}^\infty c_k x^k - \sum_{k=0}^\infty d_k x^k = \sum_{k=0}^\infty (c_k-d_k) x^k $$
Since for $x \in (-R,R)$ we have $f(x)=g(x)$ it follows that $h(x)=0$. Now we can use the fact that the coefficients of $h$ must also be zero, i.e. $c_k-d_k=0$ for all $k$ and therefore $c_k=d_k$.
Since the coefficients are all equal we have in fact that the power series are equal and therefore also their radii of convergence must be equal.
A: Suppose $f$ equals a power series convergent in $(-4,4)$ and $f=0$ in $(3,4).$ Then in the interval $(2,4),$ $f$ equals its Taylor series based at $3.$ But clearly all derivatives of $f$ at $3$ are $0.$ Thus $f=0$ in $(2,4).$ But then $f=0$ in $(0,4)$ by the same reasoning. This implies the Taylor coefficients at $0$ are all $0,$ and these are the original power series coefficients we started with. Thus $f=0$ in $(-4,4).$
