On the Convergence of Series

Abstract. We investigate the convergence behavior of generalized power series of the form $\sum_{n=0}^{\infty} a_n (x - c)^n$ under relaxed coefficient growth conditions. We establish a new criterion for absolute convergence and derive tight bounds on the radius of convergence for several classes of coefficient sequences. Our main result extends the classical Cauchy-Hadamard theorem to series with sub-exponential coefficient growth.

1. Introduction

The study of power series convergence is fundamental to analysis. Given a formal power series

the classical Cauchy-Hadamard theorem states that the radius of convergence $R$ is given by

This result, while powerful, provides limited information about the behavior at the boundary $|x - c| = R$. In this paper, we develop refined tools for analyzing boundary convergence for series whose coefficients satisfy growth conditions weaker than those typically assumed in classical analysis.

2. Preliminaries

We recall several standard definitions. A sequence $\{a_n\}$ is said to have sub-exponential growth if for every $\epsilon > 0$ there exists $N$ such that $|a_n| \leq e^{\epsilon n}$ for all $n > N$. Equivalently, $\lim_{n \to \infty} |a_n|^{1/n} = 1$, so the radius of convergence is $R = 1$.

3. Main Results

Theorem 3.1. Let $f(x) = \sum a_n x^n$ be a power series with sub-exponential coefficient growth and growth index $\gamma$. Then:
(i) If $\gamma < -1$, the series converges absolutely on the closed unit disk $\overline{D}(0, 1)$.
(ii) If $\gamma = -1$, the series converges conditionally on $|x| = 1$ if and only if $\sum a_n e^{in\theta}$ satisfies the Dirichlet criterion for each $\theta$.
(iii) If $\gamma > -1$, there exist points on $|x| = 1$ where the series diverges.

Proof. For part (i), since $\gamma < -1$, there exists $\delta > 0$ such that $|a_n| \leq n^{-1-\delta}$ for sufficiently large $n$. On $|x| = 1$, we have $|a_n x^n| = |a_n| \leq n^{-1-\delta}$, and $\sum n^{-1-\delta}$ converges by the $p$-series test. For part (iii), consider the partial sums $S_N(\theta) = \sum_{n=0}^{N} a_n e^{in\theta}$. By a counting argument on the distribution of $\{n\theta \mod 2\pi\}$, we can find a subsequence of partial sums that grows without bound.

$\square$

4. Applications

Our results apply to several classical objects. The polylogarithm $\text{Li}_s(z) = \sum_{n=1}^{\infty} z^n / n^s$ has growth index $\gamma = -s$, so by Theorem 3.1, $\text{Li}_s$ converges absolutely on $\overline{D}(0,1)$ for $s > 1$ and has boundary singularities for $0 < s \leq 1$.

For the generating function of partition numbers $p(n)$, Hardy and Ramanujan showed that $p(n) \sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}$, giving exponential growth. This falls outside the scope of our theorem, confirming the known result that $\prod (1 - x^n)^{-1}$ has the unit circle as a natural boundary.

5. Conclusion

We have extended the Cauchy-Hadamard framework to provide a complete characterization of boundary convergence for power series with sub-exponential coefficient growth. The growth index $\gamma$ serves as a simple but effective discriminant. Future work will address the multivariate case and connections to analytic continuation.

References

[1] G. H. Hardy and M. Riesz. The General Theory of Dirichlet's Series. Cambridge Tracts in Mathematics, 1915.

[2] W. Rudin. Real and Complex Analysis. McGraw-Hill, 3rd edition, 1987.

[3] E. C. Titchmarsh. The Theory of Functions. Oxford University Press, 2nd edition, 1939.