# proof of identity theorem of power series

We start by proving a more modest result. Namely, we show that, under the hypotheses of the theorem we are trying to prove, we can conclude that $a_{0}=b_{0}$.

Let $R$ be chosen such that both series converge when $|z-z_{0}|. From the set of points at which the two power series are equal, we may choose a sequence $\{w_{k}\}_{k=0}^{\infty}$ such that

• $|w_{k}-z_{0}| for all $k$.

• $\lim_{k\to\infty}w_{k}$ exists and equals $z_{0}$.

• $w_{k}\neq z_{0}$ for all $k$.

.

Since power series converge uniformly, we may interchange the limit with the summation.

 $\displaystyle\lim_{k\to\infty}\sum_{n=0}^{\infty}a_{n}(w_{k}-z_{0})^{n}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\lim_{k\to\infty}a_{n}(w_{k}-z_{0})^{n}=a_{0}$ $\displaystyle\lim_{k\to\infty}\sum_{n=0}^{\infty}b_{n}(w_{k}-z_{0})^{n}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\lim_{k\to\infty}b_{n}(w_{k}-z_{0})^{n}=b_{0}$

Because $\sum_{n=0}^{\infty}a_{n}(w_{k}-z_{0})^{n}=sum_{n=0}^{\infty}a_{n}(w_{k}-z_{0})% ^{n}$ for all $k$, this means that $a_{0}=b_{0}$.

We will now prove that $a_{n}=b_{n}$ for all $n$ by an induction argument. The intial step with $n=0$ is, of course, the result demonstrated above. Assume that $a_{m}=b_{m}$ for all $m$ less than some integer $N$. Then we have

 $\sum_{n=N}^{\infty}a_{n}(w-z_{0})^{n}=\sum_{n=N}^{\infty}b_{n}(w-z_{0})^{n}$

for all $w\in S$. Pulling out a common factor and relabelling the index, we have

 $(w-z_{0})^{N}\sum_{n=0}^{\infty}a_{n+N}(w-z_{0})^{n}=(w-z_{0})^{N}\sum_{n=0}^{% \infty}b_{n+N}(w-z_{0})^{n}.$

Because $z_{0}\notin S$, the factor $w-z_{0}$ will not equal zero, so we may cancel it:

 $\sum_{n=0}^{\infty}a_{n+N}(w-z_{0})^{n}=\sum_{n=0}^{\infty}b_{n+N}(w-z_{0})^{n}$

By our weaker result, we have $a_{N}=b_{N}$. Hence, by induction, we have $a_{n}=b_{n}$ for all $n$.

Title proof of identity theorem of power series ProofOfIdentityTheoremOfPowerSeries 2013-03-22 16:47:38 2013-03-22 16:47:38 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Proof msc 30B10 msc 40A30