Lesson 11 2 Series Vocabulary Series summation of

Lesson 11 -2 Series

Vocabulary • Series – summation of a infinite sequence ∑ s 1 + s 2 + s 3 + s 4 + …. . + sn • Partial Sum – sum of part of a infinite sequence from n=1 to n=k s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 sn = a 1 + a 2 + a 3 + …. + an • S – a real number that is the sum of the series (if it exists) • Series converges – if the sequence converges and the limit of the series as n→∞ equals a real number, s • Sequence diverges – if it does not converge

Types of Series • Geometric – ∑ arn-1 = a + ar 2 + ar 3+ …. . = a / (1 – r) |r| < 1 • Telescoping – ∑ ag(n) = a + ag(n) – ag(n) + a + …. . where third term cancels out the second and so forth until only the first and last term exist usually convergent • Harmonic – ∑ 1/n = 1 + 1/2 + 1/3 + 1/4 + …. . • and others diverges

Series Theorems ∞ an converges, then Lim an = 0 • If the series ∑ i=1 n→∞ (the converse is not true! Just because an goes to zero, doesn’t mean the series converges) • If Lim an is DNE or if lim an ≠ 0 , then the series n→∞ is divergent ∞ ∞ an and ∑ bn are convergent series, then so are • If ∑ i=1 the following series

11 -2 Example 1 Is the following series convergent or divergent? If it converges, then what does it converge to? series ½ + ¼ + 1/8 + 1/16 + …. . + 1 / 2ⁿ Lim (½)n = 0 so it might converge n→∞ Geometric Series: with a = ½ and r = ½ so it will converge to a/(1 -r) = ½ / ½ = 1

11 -2 Example 2 Is the following series convergent or divergent? If it converges, then what does it converge to? series 3/10 + 6/19 + 9/28 + 12/37 + …. . + Lim n→∞ 3 n -------9 n +1 = 1/3 so diverges 3 n -------9 n +1

11 -2 Example 3 Is the following series convergent or divergent? If it converges, then what does it converge to? series ½ + 1/3 + 1/4 + 1/5 + …. . + 1 / (n + 1) Lim 1/(n+1) = 0 so it might converge n→∞ Harmonic Series: starting at 1/2 so it will diverge

11 -2 Example 4 Examine the series below and determine if it converges or diverges; if it converges then what is it’s sum. sequence nth term is 3/n 3 Lim an = Lim ------ = 0 n n→∞ Therefore the series might converge The series can be rewritten as 3∑(1/n) which is 3 times a harmonic series! The series diverges

11 -2 Example 5 Examine the series below and determine if it converges or diverges; if it converges then what is it’s sum. series nth term is (x)n Lim an = Lim xⁿ exists only if |x| < 1 n→∞ Therefore the series might converge for |x| < 1 The series is a Geometric Series with a = 1 and r = x The series sums to 1 / (1 – x)

Homework Pg 720 – 721: Monday: problems 4, 9, 11, 15, 16 Tuesday: problems 23 -5, 35, 41