1. Given
I. an = 1
Calculus II – Worksheet 12
8.2 Intro to Series
n
II. {bn} = 1,−4, 16,−64,…
3 9 27
III. cn =(−1)ncosπ n
(a) Which sequence(s) above converge?
(b) Which of the following series converge?
∞
I. an n=1
II. 1−4 +16 −64 +··· 3 9 27
∞
III. cn n=1
2. Determine whether the series converges. For geometric or telescoping series, compute the exact sum. Write “divergent” if it diverges.
∞ 3n
(a) (b)
7
∞ 2n+1
n=2
5 · 3n+2 (c)∞ 2
n=3 ∞
(d) tann n=1
3. A ball is dropped from a height of 6ft. Each time the ball bounces, it comes back up to one-half of its previous height. What is the total distance that the ball travels?
4. Express 1.035353535 . . . as the ratio of integers.
n=0
n2 − 1
Extra Practice
5. Express 3.251515151 . . . as the ratio of integers.
6. Express 6.254254 . . . as the ratio of integers.
7. Determine whether the series is convergent or divergent by expressing SN as a telescoping sum. If it is convergent, find its sum.
∞ n (a) ln n+1
n=1 (b)∞ 2
n2 +4n+3
8. Determine whether the series converge or diverge: ∞
n=1
9. The ruler of India is very pleased with one of his palace wise men who has just invented chess. He offers the wise man a reward of his choosing. The wise man asks his master to put a grain of rice on the first square, two on the second, four on the third, and so on. How many grains does the wise man get? (note: there are 64 squares on a chess board).
n=1
n + 1 2n−3