Limit of Sum over Power Series

Calculate the limit as $n \to \infty$ of $\frac{1^a + 1 + 2^a + 1 + \ldots + n^a + 1}{n (1^a + 2^a + \ldots + n^a)}$. Determine the allowable values of $a$.

In this problem, we're tasked with evaluating the limit of a sequence as it approaches infinity, specifically focusing on the ratio involving sums of powers of natural numbers. This type of problem often requires careful application of tools from sequences and series, particularly emphasizing understanding of asymptotic behavior and convergence criteria.

One method to solve this involves approximating the sums involved using integrals, which leverages the connection between discrete sums and continuous integrals in the context of convergence. The concept of comparing the growth rates of the numerator and the denominator as the variable approaches infinity is crucial. Determining the allowable values of "a" hinges on ensuring that both the numerator and the denominator converge appropriately, which will involve insights into the behavior of power sums and potentially the use of the Riemann integral as a tool for approximation.

Furthermore, understanding when and how these series converge is a vital skill in real analysis, as it ties into broader concepts like series convergence tests and the manipulation of infinite sums. Such problems not only strengthen one's ability to manipulate inequalities and asymptotic notations but also deepen one's insight into the richness of analysis and the interplay between discrete and continuous mathematics.