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.
A 20-year 7 percent bond on a flat 10 percent curve prices at 744.5931, and a one-point rise costs exactly 63.1262. The tangent alone says 67.7028, and adding the second-order term of 4.8416 lands at 62.8612, inside 27 cents of the truth. Note that the correction and the error it corrects are two different numbers, which is why the estimate ends up on the wrong side of the answer.
Whether an up move followed by a down move lands where a down move followed by an up move lands decides between a quadratic node count and an exponential one, and at twenty steps the gap is a factor of 9,078.6. Both sums carry N+1 terms rather than N, because a twenty-step tree has twenty-one dates on it, and the off-by-one costs the entire final row. The article also states the recombination hypothesis exactly, which is weaker than the usual ud = 1.
An eight per cent thirty-year bond at par loses 27.49 dollars when its yield rises 25 basis points, and its yield moves because the principal is collateralised in United States Treasuries. The answer that circulates, about thirty-five dollars, needs a duration of fifteen, and a par bond at an eight per cent yield cannot have one: its modified duration is its own annuity factor, capped at 12.5 at any maturity whatsoever. The pass-through, the only soft number in the chain, is swept from an eighth to a half.
A three-month at-the-money call on a stock at 100 with 40% volatility is worth about eight dollars, and you can get there in two multiplications. The constant four tenths turns out to be the height of the normal bell at its peak, and the whole error of the mental rule is one rounding plus one cubic term. Scaling volatility linearly with time instead of with its square root gives ten dollars, which is 25.5% too high.
Two riders twenty-five miles apart close at fifty miles an hour, so a forty mile an hour fly shuttling between them flies exactly twenty miles. The series of shuttle legs gives the same twenty, with a first leg of 100/7 and a round-trip ratio of 1/21, but no finite number of legs ever reaches it: after eighty legs the exact total is twenty minus about 2.6e-52. The general law is wD/(a+b), and it holds identically in the rider speeds rather than by luck at 20 and 30.
One lily doubling daily covers the pond on day thirty, so eight lilies must finish in 30/8 = 3.75 days. They finish on day twenty-seven, because eight is two cubed and that slides the whole schedule exactly three days earlier. The article carries the general rule that k lilies save the floor of log base two of k, the case where five lilies save only two, and the non-overlap assumption the answer quietly rests on.