Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4

In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.

1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …

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)]

Find the sum of each series in Exercises 45–52.

∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.

aₙ = 2ⁿ 3ⁿ / 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) ]