1. Lesson 11-2 Series

2. Vocabulary Series – summation of a infinite sequence ∑ s 1 + s 2 + s 3 + s 4 + ….. + s n 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 s n = a 1 + a 2 + a 3 + …. + a n 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

3. Types of Series Geometric – ∑ ar n-1 = a + ar + ar 2 + ar 3 + ….. = a / (1 – r) |r| < 1 Telescoping – ∑ ag(n) = a + ag(n) – ag(n) + a + ….. where the 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 + ….. diverges and others

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

5. 11-2 Example 1 series ½ + ¼ + 1/8 + 1/16 + ….. + 1 / 2ⁿ Geometric Series: with a = ½ and r = ½ Is the following series convergent or divergent? If it converges, then what does it converge to? Lim (½) n = 0 so it might converge n→∞ so it will converge to a/(1-r) = ½ / ½ = 1

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

7. 11-2 Example 3 series ½ + 1/3 + 1/4 + 1/5 + ….. + 1 / (n + 1) Harmonic Series: starting at 1/2 Is the following series convergent or divergent? If it converges, then what does it converge to? Lim 1/(n+1) = 0 so it might converge n→∞ so it will diverge

8. 11-2 Example 4 sequence n th term is 3/n 3 Lim a n = Lim ------ = 0 n Therefore the series might converge n→∞ Examine the series below and determine if it converges or diverges; if it converges then what is it’s sum. The series diverges The series can be rewritten as 3∑(1/n) which is 3 times a harmonic series!

9. 11-2 Example 5 series n th term is (x) n Lim a n = Lim xⁿ exists only if |x| < 1 Therefore the series might converge for |x| < 1 n→∞ Examine the series below and determine if it converges or diverges; if it converges then what is it’s sum. The series sums to 1 / (1 – x) The series is a Geometric Series with a = 1 and r = x

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