advertisement

Fall 2010 Math 152 4. Compute the sum ∞ n X 1 n=0 2 n 2 or + 3 show the series diverges. Week in Review VIII 5. Write the repeating decimal 0.51 as a fraction. courtesy: David J. Manuel (covering 10.2, 10.3) 2 1 Section 10.3 Section 10.2 Key Concepts: Key Concepts: 1. Infinite Series: ∞ X an . Define sN = n=1 The series converges to s if and only s. N X n=1 if sN 1. Integral Test: If an = f (n), where f isX continuous, positive, and decreasing, then an ˆ ∞ is convergent if and only if f (x) dx con1 verges an . → 2. Remainder for Integral Test: X If an converges to s, then ˆ ∞ ˆ ∞ f (x) dx ≤ s − sN ≤ f (x) dx. 2. Series we can sum: (a) Geometric: ∞ X arn = n=1 if |r| < 1. (b) Telescoping: ∞ X a if and only 1−r N +1 (bn+1 − bn ) 4. Comparison Test: if 0 ≤ an ≤ bn , then: X X (a) If bn is convergent, then an is convergent X X (b) If an is divergent, then bn is divergent n=1 3. Test for Divergence: If then an → 0. X N X 1 3. P-Test: is convergent if and only if np p > 1. an is convergent, Examples: 5. Limit Comparison Test: If anX > 0 and bX n > 0 an and 0 < lim < ∞, then an and bn n→∞ bn either both converge or both diverge. ∞ X 3 1. Given the series , find a formula 2+n n n=1 for sN and determine if the series converges or diverges. If it converges, find the sum. 2. Given a series whose partial sums are given by Examples: sN = ln(N + 1), find an and determine if the series converges or diverges. If it converges, 1. Determine whether the following series confind the sum. verge or diverge. State the name of the test used and show all conditions are met. 3. A student begins running laps on a 440 yard track. He finishes the first lap in 1 minute, but finds that for each subsequent minute he 9 the distance of the previous only covers 10 minute. If the student could run indefinitely, what distance will he cover? 1 ∞ X (a) 1 1 + n2 n=0 (b) ∞ X 1 1/n 2 n=1 (c) (d) (e) (f) ∞ X n=1 ∞ X 2n 1 +n 1 √ 3 n +2 n=1 ∞ X n=1 ∞ X n=1 n2 2n √ − n+1 sin π n 2. The partial sum s10 of ∞ X 2 ne−n is 0.40488. n=1 Find an upper and lower bound on the sum of the series. 3. How many terms of the series are needed to ∞ X 1 approximate to within 2 × 10−9 ? 5 n n=1 2