Why limes superior in Cauchy-Hadamard formula for radius of convergence of power series?

Question

Can anyone explain to me why there is $\limsup$ instead of $\lim$ in Cauchy-Hadamard formula for radius of convergence of power series? It isn't that obvious to me ;/

Answer 1

To understand this, consider a series $\sum\limits_na_nx^n$ when $(|a_n|^{1/n})$ has no limit, say $a_{2n}=2^{2n}$ and $a_{2n+1}=3^{2n+1}$ for every $n$. Then $\limsup|a_n|^{1/n}=3$ and $\liminf|a_n|^{1/n}=2$ hence $(|a_n|^{1/n})$ diverges. The sequence $(a_nz^n)$ is unbounded when $|z|\gt1/3$ and $|a_nz^n|\leqslant(r_z)^n$ when $|z|\lt1/3$ for some $r_z\lt1$ that I will let you discover, hence the radius of convergence is indeed $1/3$.

Answer 2

The idea is you only need an upper bound for "the end" of the sequence of coefficients. Whether or not the coefficients converge (that is, lim inf = lim sup) is irrelevant to whether the power series converges.

For $\sum c_n x^n$ to converge for certain $x$, we only need $|c_n|$ to be small in some sense. Assuming $|c_n|$ is bigger than something doesn't help convergence.