Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

#18 Wednesday (10/23/96)

SUMMARY Infinite Series READING §14 (pages 68-75) \underline{Warm-Up} (5-10 minutes) Recall all that you know about infinite series. Write down an example of one. What notation are you using? How do you determine convergence? Etc.. \underline{Definition} Infinite series

s_{n} = ∑_{k=m}^{n} a_{k} =

s_{n} is called a 14.2 An infinite series is said to converge to a number S when

∑_{n=m}^{∞} a_{n} =

A series that does not converge is said to If the terms a_{n} of an infinite series are nonnegative (≥ 0) then we know that (s_{n}) is a

sequence.

The series ∑ a_{n} is said to converge absolutely if ∑ |a_{n}| converges The series ∑(−1)^{n}a_{n} is said to be an alternating series if a_{n} ≥ 0 (or a_{n} ≤ 0) for all b ∈ ℕ

CONVERGENCE OF INFINITE SERIES

Geometric Series Try and recall formulas about the convergence of geometric series. ∑_{n=0}^{∞} ar^{n} =

If |r| < 1 then ∑_{n=0}^{∞} ar^{n} =

\underline{p-Series} ∑_{n=1}^{∞} \frac{1}{n^{p}} CONVERGES if and only if p > 1

(14.6) Comparison Test Let ∑ a_{n} be a series with a_{n} ≥ 0∀n

(i) If ∑ a_{n} converges and |b_{n}| ≤ a_{n} for all n, then ∑ b_{n} converges (ii) If ∑ a_{n} = +∞ and b_{n} ≥ a_{n} for all n, then ∑ b_{n} = +∞

(14.8) Ratio Test A series ∑ a_{n} of nonzero terms

(i) converges absolutely if lim sup |a_{n+1}/a_{n}| < 1 (ii) diverges if lim sup |a_{n+1}/a_{n}| > 1 (iii) Otherwise if lim inf |a_{n+1}/a_{n}| ≤ 1 ≤ lim sup |a_{n+1}/a_{n}| and the test gives no information

(14.9) Root Test Let ∑ a_{n} be a series and let α = lim sup |a_{n}|^{1/n}. The series ∑ a_{n}

(i) converges absolutely if α < 1 (ii) diverges if α > 1 (iii) Otherwise if α = 1 and the test gives no information

Homework Assignment (Due Friday October 25) §11, page 56: 11.4, 11.5 §12, page 60: 12.3, 12.14(a)