In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (ln n) xⁿ ]

Finding Taylor Series

Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.

e⁻ˣ/²

Use series to evaluate the limits in Exercises 29–40.

29. lim (x → 0) (e^x - (1 + x)) / x²

If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (tanh n) / n²

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) cos²n / n^(3/2)