Examples of Comparison Example: ∞ � 1 √ 2 n +1 n=0 We know that √n12 +1 is comparable to √1n2 = n1 , so by limit comparison we know �∞ �∞ that n=1 √n12 +1 converges or diverges as n=0 n1 does. We proved earlier that �∞ 1 �∞ √ 1 n=1 n diverges, so n=0 n2 +1 must diverge as well. �∞ �∞ Note that we can include the n = 0 term in n=0 √n12 +1 but not in n=1 n1 . This is ok; the limit comparison test is is concerned only with long term behavior, not with the early partial sums. Example: ∞ � 1 √ 5 n − n2 n=2 Because we have a subtraction in the denominator we have to be careful not to divide by zero; we start our series at n = 2. We compare √n51−n2 to √1n5 = n51/2 . If we choose the right function to be the numerator, it’s relatively simple to show that they are similar: f (n) n→∞ g(n) lim = √1 n5 n→∞ √ 1 n5 −n2 lim = √ n 5 − n2 √ n→∞ n5 � n5 n2 lim − 5 5 n→∞ n n √ 1−0 = 1. = = lim Since the two functions are we can apply the limit �∞similar, �∞comparison test 1 and conclude that because n=1 n5/2 converges ( 52 > 1), n=2 √n51−n2 must also converge.

Examples of Comparison

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock