Reference no: EM132588519

Math 25 Advanced Calculus

Problem 1. Prove that the sequence

a_n = 1 · 3 · 5 · . . . · (2n - 1) /( 2 · 4 · 6 · . . . · (2n))

converges. (Note: you are not asked to find what the limit of the sequence is.)

Problem 2. Define a sequence (x_n)^∞_n=1 by
x₁ = √3, x₂ = √(3 + √3), x₃ = √(3 + √(3 + √3)), . . . ,

x_n+1 = √(3 + x_n), . . .

Prove that the sequence (x_n)^∞_n=1 converges, and find its limit. For a small bonus credit, answer the same question for the more general sequence (y_n)^∞_n=1 defined by

y₁ = √k, y_n+1 = √(k + y_n),

where k is an arbitary integer 2 (note that in that case the limit is a function of k).

Problem 3. Define the sequence (a_n)^∞_n=1 by a_n = (1 + 1/n)ⁿ .

(a) Show that (an) is increasing and bounded from above (see Example 2.36 in section 2.10 of the textbook for ideas) and deduce that it converges.

Define the constant e by e = lim a_n.
                                       n→∞

(b) Show that 2 < e < 3.

Problem 4. In an analog clock, at twelve o'clock both the hours dial and the minutes dial are pointing in the same direction. In this problem, we will find the next time that the two dials are aligned, in two different ways.

(a) Let X be the number of minutes until the two dials point in the same direction again. Write an algebraic equation that X satisfies, and solve it to find X.

(b) Another way to compute X is by representing it as the sum of an infinite geometric series, as follows. Define a sequence of times T0, T1, T2, T3, . . . by
T0 = twelve o'clock,
T1 = the first time after twelve o'clock when the minutes dial returns to the position where the hours dial was at twelve o'clock,
T2 = the first time after time T1 when the minutes dial arrives
at the position where the hours dial was at time T1,
.
Tn = the first time after time Tn-1 when the minutes dial arrives at the position where the hours dial was at time Tn-1, etc. For example, it is easy to see that T1 = 1:00 a.m. and T2 = 1:05 a.m. Explain why Tn can be written in the form Tn = twelve o'clock + x1 + x2 + . . . + xn,
where xn is the time interval between Tn-1 and Tn. Find a formula for xn representing it as a geometric progression. Deduce that
X = lim (x1 + x2 + x3 + . . . + xn),
      n→∞
and use the formula
1 + a + a² + a³ + . . . = 1/(1-a) , (|a| < 1)

for the sum of an infinite geometric series, to compute X in another way.

(c) Finally, compare the answers you got for X in parts (a) and (b). If they are not the same, you made a mistake somewhere - check your solution and correct it.