ANALYSIS I: PROBLEM SET # 4
DUE FRIDAY, 11 MARCH
Exercise 29. Fix an integer b , and suppose α a real number. Set x0 := bαc, and define a sequence (xn )n≥0 recursively
in the following manner. For any integer n ≥ 1, set
xn := xn−1 +
b(α − xn−1 )b n c
bn
Does the sequence (xn )n≥0 converge? If so, to what does it converge?
.
Exercise 30. Consider the sequence (xn )n≥1 of real numbers defined by
1 n
:=
.
xn
1+
n
Show that (xn )n≥1 is a Cauchy sequence. Its limit is called Euler’s constant, denoted e. Compute this value to three
decimal places.
Exercise 31. Suppose x0 , x1 ∈ R distinct real numbers. Define a sequence (xn )n≥0 recursively in the following
manner. For any integer n ≥ 2, let xn := 12 (xn−1 + xn−2 ). Show that (xn )n≥0 is a Cauchy sequence.
Exercise 32. Set x0 := 3. Define a sequence (xn )n≥0 recursively in the following manner. For any integer n ≥ 1, set
xn := xn−1 −
(−1)n 4
(2n)(2n + 1)(2n + 2)
.
Show that the sequence (xn )n≥0 is Cauchy. Guess its limit.
Exercise 33. Suppose N a positive integer. Suppose x0 any nonnegative real number. Now define a sequence (xn )n≥0
recursively in the following manner. For any integer n ≥ 1, set
‚
Œ
N
1
xn :=
xn−1 +
.
2
xn−1
Does the sequence (xn )n≥0 converge? If so, to what does it converge?
1