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Problem 10.1.73d
72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence
d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.
Radioactive decay
A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.
Problem 10.2.83d
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
d. If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and
{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},
then limₙ→∞ aₙ = limₙ→∞ bₙ.
Problem 10.3.87d
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.
Problem 10.3.87e
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.
Problem 10.4.47e
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.
Problem 10.2.83f
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
f. If the sequence {aₙ} diverges, then the sequence {0.000001 aₙ} diverges.
Problem 10.4.47f
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."
Problem 10.3.87f
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.
Problem 10.3.87g
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
g. Viewed as a function of r, the series 1 + r + r² + r³ + ⋯ takes on all values in the interval (1/2, ∞).