Infinite Series Convergence

This question focuses on the convergence of an infinite series whose terms decay extremely fast, faster than any power of 1/n. The setup is a standard calculus problem about determining whether a sum of positive terms defined via an exponential expression in n converges or diverges. Candidates must reason about the long-run behavior of the terms and relate it to benchmark series they already know from basic analysis, such as p-series or geometric series. It sits in the usual toolkit of questions used in university-level calculus and real analysis courses to probe understanding of infinite series.

To tackle it, a candidate needs to recognize and exploit exponential decay and compare it to more familiar polynomial decay. The most natural approach involves either the Ratio Test or a direct comparison to a geometric or p-series, carefully tracking how the terms behave as n grows. An interviewer is watching for clear justification of test selection, correct limit computations, and the ability to articulate why the decay rate is decisive for convergence rather than relying on intuition or hand-waving.

What it tests

For series whose terms decay exponentially or faster (such as $e^{-n^k}$ for $k > 1$), the convergence is governed by how rapidly the terms approach zero relative to familiar benchmark series. The key insight is that exponential decay outpaces any polynomial decay: as $n$ increases, $e^{-n^2}$ shrinks much faster than $1/n^p$ for any $p > 0$. This means that if a series with polynomially decaying terms converges (like the p-series for $p > 1$), then a series with exponentially decaying terms will also converge, and often much more rapidly. The Ratio Test is especially powerful here, as it quantifies the rate at which terms decrease and provides a clear threshold for convergence: if the ratio of successive terms tends to a limit less than one, the series converges. This principle holds because the exponential function's growth (or decay) dominates polynomials for large arguments, making exponential series 'safer' for convergence.