발산 판정법과 적분 판정법

In certain applications of probability, such as the so-called Watterson estimator for predicting mutation rates in population genetics, it is important to have an accurate estimate of the number ${H}_{k}=\left(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{k}\right).$ Recall that ${T}_{k}={H}_{k}-\ln k$ is decreasing. Compute $T=\lim_{k\to \infty }{T}_{k}$ to four decimal places. (Hint: $\frac{1}{k+1}<{\int }_{k}^{k+1}\frac{1}{x}dx$.)

Suppose a scooter can travel $100$ km on a full tank of fuel. Assuming that fuel can be transferred from one scooter to another but can only be carried in the tank, present a procedure that will enable one of the scooters to travel $100{H}_{N}$ km, where ${H}_{N}=1+1/2+\cdots +1/N.$

[T] Suppose a computer can sum one million terms per second of the divergent series $\sum_{n=1}^{N}\frac{1}{n}.$ Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed $100.$

[T] A fast computer can sum one million terms per second of the divergent series $\sum_{n=2}^{N}\frac{1}{n\ln n}.$ Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed $100.$