27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
a. Find the next two terms of the sequence.
{1, 2, 4, 8, 16, ......}

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯

89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.

a. Find the base-10 representation of the binary number 0.001̅1.

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

{-5, 5, -5, 5, ......}

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

a. For what values of p does this series converge?

{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …

a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.