Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]
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 ∞) (1 − n) / n2ⁿ
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 ∞) [ (x − 1)ⁿ / (n³ 3ⁿ) ]
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]
Use series to evaluate the limits in Exercises 29–40.
37. lim (x → 0) ln(1 + x²) / (1 - cos(x))
