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 − …

Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.

Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = √(1 − x),a = 0

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]

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!

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