Nth term & Alternating Series

1. Nth term & Alternating Series Dean Lee & Jenny Park AP Calculus BC G Block 9/2/2011

2. Practice Problems • Does the series converge? 1. 2. (Harmonic Series) 3.

3. Nth term Test • Requirement for convergence of a series • nth term approaches 0 as n -> infinity • The nth term test for divergence: • n=∞1∑an diverges if (limn->∞ an) fails to exist or is different from 0

4. Answers

5. Alternating Series Test • The series n=∞1∑(-1)n+1un = u1 - u2 + u3 - u4 + … converges if all three of the following conditions are satisfied: • 1. Each un is positive • 2. un ≥ un+1 for all n ≥ N, for some integer N • 3. lim n->∞ un ->0

6. Alternating Series Examples • Problem: • Solution: • limn->∞ an = limn->∞ (1/n) = 0 • ∴ the third condition is satisfied • an+1 = 1/(n+1) ≤ 1/n = an, for all n • ∴ the second condition is satisfied • Each un is positive, since we start from n=1 and begin with a positive term • ∴ the first condition is satisfied • Thus, the series converges!

7. Alternating Series Test • Problem: • Solution: • limn->∞ an = limn->∞ (n/(-2)n-1) = 0 • ∴ the third condition is satisfied • ∴ the second condition is satisfied • Each un is positive, since we start from n=1 and begin with a positive term • ∴ the first condition is satisfied • Thus, the series converges!

8. Exceptions • passes the second condition of the Alternating Series Test, but the series does not pass the third condition. In fact, the series diverges. • passes the third condition, but you cannot apply the Alternating Series Test because the series does not pass the second condition. To conclude that the series diverges, you can argue that S2N equals the Nth partial sum of the divergent harmonic series. This implies that the sequence of partial sums diverges. So, the series diverges.

9. Bibliography • Larson, Ron, and Bruce H. Edwards. Calculus. Belmont, CA: Brooks/Cole, Cengage Learning, 2010. Print. • Finney, Damana, Waits, and Kennedy. Calculus: Graphical, Numerical, Algebraic. Upper Saddle River, NJ: Prentice Hall, Pearson Education, Inc. 2003. Print.