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Unformatted text preview: * Know how to find the N th partial sum of simple series (such as telescoping series) and use it to compute the sum of the series. * If  r  < 1, then the geometric series ∞ X n =0 ar n = a + ar + ar 2 + ··· + ar n + ··· coverges and its sum is a 1 r ; that is, ∞ X n =0 ar n = a + ar + ar 2 + ··· + ar n + ··· = a 1 r The series diverges if  r  ≥ 1. * Warning, the above fact talks about geometric series beginning at n = 0. If your geometric series does not begin at n = 0, you need to manipulate it so that it does. Here’s how: ∞ X n = k ar n = ∞ X n =0 ( ar k ) r n * Properties of Infinite Series: If ∞ X n =1 a n and ∞ X n =1 b n are convergent infinite series and c is a constant, then 1. ∞ X n =1 ca n = c ∞ X n =1 a n 2. ∞ X n =1 ( a n ± b n ) = ∞ X n =1 a n ± ∞ X n =1 b n * Exercises: pp.718719 134 – 11.4 Series with Positive Terms * The Test for Divergence: If lim n →∞ a n does not exist or lim n →∞ a n 6 = 0, then ∞ X n =1 a n diverges. * Warning: It is NOT true in general that lim n →∞ a n = 0 implies that ∑ ∞ n =1 a n converges. Here is a counterexample, the harmonic series ∞ X n =1 1 n does not converge but lim n →∞ 1 n = 0. 11 * The Integral Test: Suppose f is a continuous, positive, decreasing function on [1 , ∞ ). If f ( n ) = a n for all n ≥ 1 then ∞ X n =1 a n and Z ∞ 1 f ( x ) dx either both converge or both diverge. Note: there are several assumptions you need to verify are met before you can use the integral test. * Convergence of pSeries: The pseries ∞ X n =1 1 n p converges if p > 1 and diverges if p ≤ 1. * The Comparison Test: Suppose ∑ a n and ∑ b n are series with positive terms. 1. If ∑ b n is convergent and a n ≤ b n , for all n , then ∑ a n is also convergent. 2. If ∑ b n is divergent and a n ≥ b n , for all n , then ∑ a n is also divergent. * When using the comparison test, you need to verify that both series being compared are seris with positive terms. * Exercises: p.729 148 – 11.5 Power Series and Taylor Series * Terms: Taylor series, power series, interval of convergence, radius of convergence * Suppose that we are given the power series ∞ X n =0 a n ( x a ) n Let R = lim n →∞  a n a n +1  1. If R = 0, the series converges only for x = a . 2. If 0 < R < ∞ , the series converges for x in the interval ( a R,a + R ) and diverges for x outside the interval [ a R,a + R ] 3. If R = ∞ , the series converges for all x . * Suppose the function f is defined by f ( x ) = ∞ X n =0 a n ( x a ) n = a + a 1 ( x 1) + a 2 ( x 2) 2 + ··· with radius of convergence R > 0. Then f ( x ) = ∞ X n =1 na n ( x a ) n 1 = a 1 + 2 a 2 ( x a ) + 3 a 3 ( x a ) 2 + ··· on the interval ( a R,a + R )....
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 Spring '12
 TA
 Math, Normal Distribution, Probability distribution, Probability theory, probability density function, dx

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