Chapter 11 Theorems - Lecture notes 3 PDF

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Chapter 11 Theorems Sequences: 1. Remember Chapter 2 2. L’Hospital’s Rule: In case of indeterminate quotient 𝑓 ′(𝑥)

∞ 0 , ∞ 0

:

𝑓(𝑥) = lim ′ 𝑥→𝑎 𝑔 (𝑥) 𝑥→𝑎 𝑔(𝑥) 3. Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent. 4. Squeeze Theorem lim

Series: 1. Geometric Series: ∞

∑ 𝑎𝑟 𝑛−1 =

𝑛=1

𝑎 1−𝑟

converges for |𝑟| < 1 diverges for 𝑟 ≥ 1

2. Test for Divergence: If lim 𝑎𝑛 ≠ 0, then the series ∑ ∞ 𝑛=1 𝑎𝑛 is divergent 𝑛→∞

3. Harmonic Series: ∞

∑

1

𝑛=1

𝑛

𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠

4. P-Series: ∞

∑

1 𝑛𝑝

𝑛=1

convergent if 𝑝 > 1 divergent if 𝑝 ≤ 1 5. Integral Test : Suppose 𝑓 is a continuous, positive, decreasing function on [1, ∞) and let 𝑎𝑛 = 𝑓(𝑛). Then ∞

a. If ∫1 𝑓(𝑥) 𝑑𝑥 is convergent, then ∑∞ 𝑛=1 𝑎𝑛 is convergent ∞

b. If ∫1 𝑓(𝑥) 𝑑𝑥 is divergent, then ∑∞ 𝑛=1 𝑎𝑛 is divergent

6. The Comparison Test: Suppose that ∑ 𝑎𝑛 and ∑ 𝑏𝑛 are series with positive terms a. If ∑ 𝑏𝑛 is convergent and 𝑎𝑛 ≤ 𝑏𝑛 for all n, then ∑ 𝑎𝑛 is also convergent b. If ∑ 𝑏𝑛 is divergent and 𝑎𝑛 ≥ 𝑏𝑛 for all n, then ∑ 𝑎𝑛 is also divergent 7. The Limit Comparison Test: Suppose that ∑ 𝑎𝑛 and ∑ 𝑏𝑛 are series with positive terms. If 𝑎𝑛 =𝑐 lim 𝑛→∞ 𝑏𝑛 where 𝑐 is a finite number and 𝑐 > 0, then either both series converge or both diverge. 8. Alternating Series Test: 𝑛−1 𝑏 satisfies the following two conditions If the alternating series ∑∞ 𝑛=1 (−1) 𝑛 a. 𝑏𝑛+1 ≤ 𝑏𝑛 for all n b. lim 𝑏𝑛 = 0 𝑛→∞

then the series is convergent. 9. Alternating Series Estimation Theorem: If the alternating series ∑(−1)𝑛−1 𝑏𝑛 is convergent, then |𝑅𝑛 | = |𝑠 − 𝑠𝑛 | ≤ 𝑏𝑛+1 10. The Ratio Test: a. If lim | 𝑛→∞

b. If lim | 𝑛→∞

c. If lim | 𝑛→∞

𝑎𝑛+1 | 𝑎𝑛 𝑎𝑛+1

𝑎𝑛 𝑎𝑛+1 𝑎𝑛

= 𝐿 < 1, then the series ∑ 𝑎𝑛 is absolutely convergent

| = 𝐿 > 1, then the series ∑ 𝑎𝑛 is divergent

| = 1, the Ratio Test is inconclusive

11. The Root Test a. If lim √|𝑎𝑛 | = 𝐿 < 1, then the series ∑ 𝑎𝑛 is absolutely convergent 𝑛→∞

𝑛

𝑛 b. If lim √|𝑎𝑛 | = 𝐿 > 1, then the series ∑ 𝑎𝑛 is divergent

𝑛→∞

𝑛

c. If lim √|𝑎𝑛 | = 1, then the Root Test is inconclusive 𝑛→∞...