Theory and Examples
Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:
i) a₁ ≥ a₂ ≥ a₃ ≥ …;
ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.
Show that the series
a₁/1 + a₂/2 + a₃/3 + …
diverges.
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) (csch n)xⁿ
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 1 + (0.9)ⁿ
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (-4)ⁿ/n!
Maclaurin Series
Find Taylor series at x = 0 for the functions in Exercises 63–70.
cos (x³/√5)
