Exam 1 Review

Chapter 1

• Let a = 0.5714. Find the largest n such that |5a - 4| < 10⁻ⁿ. Find the largest n such that |7a - 4| < 10⁻ⁿ.

• State the arithmetic-geometric mean inequality.

• Show it is not possible to write √2 in the form p/q where p and q are integers.

• Write the repeating decimal 5.23 as a fraction of the form p/q where p and q are integers.

• Define in terms of N and ε what is means for lim_{n → ∞} a_n = L.

• Define what it means for a_n to be a Cauchy sequence.

• Show that every Cauchy seqence is bounded.

• Show that the harmonic series diverges.

• Sum the infinite series
  1/2⁵ + 1/2⁶ + 1/2⁷ + 1/2⁸ + …

• Sum the infinite series
  1/2⁵ − 1/2⁶ + 1/2⁷ − 1/2⁸ + …

• Explain why the series

  ∞	n2
  ∑
  n=1	n2 + 1

  diverges.

• Determine whether the series

  ∞	(-1)n n
  ∑
  n=1	√n³ + 1

  converges absolutely, coverges conditionally or diverges.

• Suppose for all n ≥ N that
  a_n ≤ b_n ≤ c_n

  and that lim_{n → ∞} a_n = lim_{n → ∞} c_n = a. Prove that lim_{n → ∞} b_n = a.

Chapter 2

• Define in terms of δ and ε what it means for a function f(x) to be continuous at c.

• Use δ and ε to show the following functions are continuous at c = 1/2
  1/x,   x²,   3x,   √x,   x³,   1/√x,   1/x²

• Suppose f(x) and g(x) are continuous at c and that g(c) ≠ 0. Use δ and ε to show that
  5f(x),   f(x)+g(x),   f(x)−g(x),   f(x)g(x),   1/g(x)

  are continuous at c.

• Suppose f(x) is decreasing and continuous.
  • Show that f⁻¹(y) is decreasing.
  • Show that f⁻¹(y) is continuous at c = f(a).

• Suppose a = g(c), that g(x) is continuous at c and that f(x) is continuous at a. Show that f(g(x)) is continuous at c.