Math 121A: Homework 1

1. Two points P and Q are connected by a straight road of length d. At time t = 0, a car starts driving from P to Q at a constant speed c. At t = 0, a bee starts flying from Q toward the car at a constant speed b > c, and takes a zig-zagging path, reversing direction each time it meets the car or point Q. This process stops when the car reaches Q. Let a1 be the distance the bee covers from Q to the car, a2 be the distance it covers from the car back to Q, a3 be the distance it covers from Q to meet the car the second time, and so on.

(a) Calculate the values of the infinite sequence a₁, a₂, a₃, . . . of distances that the bee covers.

(b) What is the total distance D = Σa_i covered by the bee? Why should this answer be expected?

(c) Optional for the enthusiasts. Suppose the bee travels at a different speed b' on the sections when it is moving from Q toward the car. What are the values of a_i and D in this case?

2. Assume that Σ1/n diverges. By using the comparison test with an appropriate series, determine whether _n=1Σ^∞1/(n + ½) converges or diverges.

3. (a) Find the exact interval of convergence of the power series

_n=1Σ^∞ xⁿ/n.

(b) Show that the power series

_n=1Σ^∞ 1/n(y/1 + y²)ⁿ, converges for all real numbers y.