Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]

Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if

b. a = 13/2

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (xⁿ / (2n + 1))^(1/n),x > 0

Make up an infinite series of nonzero terms whose sum is

b. −3

Repeating Decimals

Express each of the numbers in Exercises 23–30 as the ratio of two integers.

3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...

Finding a Sequence’s Formula

In Exercises 13–30, find a formula for the nth term of the sequence.

1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs