Make up an infinite series of nonzero terms whose sum is
b. −3

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

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

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)

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