Back54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)
Does a geometric series always have a finite value?
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.
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.
87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.
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, ∞).
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.
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
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.
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."
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.
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)
Is it possible for an alternating series to converge absolutely but not conditionally?
Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) 1 / ( (3k + 1)(3k + 4) )