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Section 10.3: The Integral and Comparison Tests The Integral Test: If f (x) is a positive, continuous, decreasing function on [1, ∞], and ak = f (k). Z Then ∞ P ∞ ak and k=1 f (x) dx either both converge or both diverge. Why is this so? 1 EXAMPLE 1: Determine whether the following series converge or diverge. a.) b.) n n=1 n + 1 ∞ P ∞ P n=1 c.) ∞ P n2 n +1 ne−n 2 n=0 d.) 1 n=2 n ln n ∞ P EXAMPLE 2: For what value(s) of p does the series 1 converge? p n=1 n ∞ P 1 1 1 = 1 + p + p + ..., where p > 0. The p 2 3 n=1 n special case where p = 1 is called the harmonic series: P-series Test: A p-series is a series of the form The p series test says ∞ P 1 is convergent if p > 1 and divergent if p ≤ 1. n=1 np ∞ P EXAMPLE 3: Determine whether the series 1000 √ converges or diverges. n=1 n n EXAMPLE 4: Determine whether the series 3 √ converges or diverges. n n=1 Note that if P ∞ P an is similar to a p series, then we may want to compare the series to a p series. The Comparison Test: Suppose ∞ P n=1 • If ∞ P an and ∞ P bn are series of positive terms. i=1 ∞ P bn is convergent and an ≤ bn for all n, then ∞ P bn is divergent and an ≥ bn for all n, then n=1 the larger series converges, so does the smaller series”. • If n=1 smaller series diverges, so does the larger series”. ∞ P n=1 ∞ P an is also convergent. In other words, ”if an is also divergent. In other words, ”if the n=1 Note: if the larger series diverges, no conclusion can be made about the smaller series. Likewise, if the smaller series converges no conclusion can be made about the larger series. EXAMPLE 5: Determine whether the following series converge or diverge. ∞ n4 P a.) 8 2 n=1 n + n + 1 b.) ∞ P n=17 √ n2 n5 − 4n c.) sin2 n √ n=1 n n d.) cos2 n + 5 √ 3 n n=1 n + e.) 5 + sin n √ n n=1 ∞ P ∞ P ∞ P The Limit Comparison Test: If the Comparison Test is inconclusive, then we may apply the Limit Comparison Test: Suppose ∞ P an and ∞ P bn are series of positive terms. n=1 n=1 an = c > 0, then either both series converge or both diverge. bn an an = 0 or lim = ∞, then the test fails and therefore we need to apply another Note: If lim n→∞ b n→∞ b n n test. If n→∞ lim EXAMPLE 6: Determine whether the following series converge or diverge. ∞ n2 − 5n P a.) n=1 n3 + 2n 1 sin 2 b.) n n=1 ∞ P c.) sin ∞ P cos n=1 d.) n=1 1 n ∞ P 1 n Remainder Estimate: Suppose sn = n P ai = a1 + a2 + ... + an is the nth partial sum of the i=1 convergent series ∞ P n=1 an . Then the remainder in using sn to approximate the sum S is defined to n=1 ∞ P be Rn = S − sn = Moreover, if ∞ P ai = an+1 + an+2 + .... i=n+1 an was shown to be convergent by the integral test where an = f (n), then Rn = ∞ X i=n+1 ai < Z ∞ n f (x) dx. 1 2 n=1 n + 4 a.) Find the sum of the first 10 terms. EXAMPLE 7: Consider ∞ P b.) Estimate the error in using the sum of the first 10 terms as an approximation to c.) Find ∞ P n=1 n2 n=1 n2 1 . +4 1 correct to 10 decimal places. +4 EXAMPLE 8: How many terms of the series 1 ? 81 ∞ P 1 would you need to add to find its sum to within n=1 n4 ∞ P 3 + cos n n5 n=1 a.) Prove the series converges. EXAMPLE 10: Consider b.) Approximate ∞ P 3 + cos n using s6 . n5 n=1 ∞ P c.) Estimate the error in using s6 to approximate 3 + cos n . n5 n=1 ∞ P