Convergence of a power series: general question

Question

Let $\sum_{n\ge 0} a_nz^n$ be a power series with $a_i\in\mathbb{C}$.

Fact 1. The limit $\lim\sup |a_n|^{1/n}$ always exists. If it is finite and non-zero, its reciprocal is the radicus of convergence of the series.

Fact 2. If the limit $\lim_{n\rightarrow \infty} |a_{n+1}|/|a_n|$ exists and non-zero, then its reciprocal is radius of convergence.

Question: Let $\sum a_nz^n$ be a power series with following conditions:

• all $a_i$ are non-zero.

• The radius of convergence is finite and non-zero (say $R=2$).

Is it necessary that $\lim_{n\rightarrow \infty} |a_{n+1}|/|a_n|$ exists?

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This may be obvious/silly question; but I came across it with example of $\sum z^{n!}$. Although many coefficients are zero, and the limit of rations of coefficients not exists, the radius of convergence is $1$; I am simply considering a case where all coefficients are non-zero.

Answer 1

Take $$f(x)=\dfrac{1}{1-x^2}+\dfrac{1}{1-x}=2+x+2x^2+x^3+2x^4+x^5+2x^6+...\qquad ,\qquad |x|<1$$ therefore $$a_n=2 \qquad ,\qquad\text{n is even}\\a_n=1 \qquad ,\qquad\text{n is odd}$$so the limit doesn't exist and the radius of convergence is $1$.

Answer 2

Take $a_n=n$ if $n\not=k!$ for all $k$ and $a_n=1$ for $n=k!$. For the corresponding series the sequence of $\vert a_{n+1}/a_n\vert$ has subsequences converging to 0 and $\infty$ respectively.