Both Term Tests Return One, and the Sum Still Stops Below Two
The ratio and root tests both return 1 on the sum of e to the minus root n, which settles nothing, and the usual write-up of the problem then quotes 4 over e as the answer. That number is the floor rather than the cap: the sum is 1.6704068, which is 13.52 percent above it, and the usable bound comes from integrating from 0 instead of from 1, giving exactly 2. Where each bar of width one sits relative to its index is the single step that decides which way the inequality points.