Sequence of polynomials which converge uniformaly to 1/z

Question

Does not there a sequence of polynomials which converges uniformly to $1/z$ on $\{z\in\Bbb C\;:\;|z|=1\}$? Explain.

Here is where I'm having problems with the question. I understand what a sequence is, however I don't know what a sequence of polynomials is. Does this mean that a you come up with a generic sequence that represents a polynomial and all these polynomials reduce down to $1/z$?

There is also uniform convergence in real analysis. Is this the same type of uniform converge?

My class mate that this the answer comes from some type of approximation of polynomials theorem.

Answer 1

Call $(p_n)_n$ such a sequence of polynomials.

Then, since polynomials are (entire and thus) holomorphic on $\gamma=\{|z|=1\}$ you should have \begin{align*} 2\pi i&=\int_{\gamma}\frac1z\,dz\\ &=\int_{\gamma}\lim_np_n(z)\,dz\\ &=\lim_n\underbrace{\int_{\gamma}p_n(z)\,dz}_{=0\;\;\forall n}=0 \end{align*} which is absurd.

Answer 2

If you restrict your interval, you can.

For example, if $|x| < 1$, then $\sum x^n \to \frac1{1-x}$.

Put $1-x$ for $x$ and $\sum (1-x)^n \to \frac1{x}$ when $|1-x|<1$.