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Problem 10.4.47d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².
Problem 10.4.41d
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.
41. ∑ (k = 1 to ∞) 1 / k⁶
Problem 10.7.31d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The Ratio Test is always inconclusive when applied to ∑ aₖ, where aₖ is a nonzero rational function of k.
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 10.4.47f
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 10.2.83f
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 10.3.87f
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, ∞).