11.4.misc1-eg - n n 3 (or 1 n 2 ) and that the series is...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Example Use the Direct Comparison Test to determine if the series converges or diverges. X n =1 ln n ( n 3 + 1) . To use the Direct Comparison test to show convergence, we need to find a convergent series X n =1 b n for which ln n ( n 3 + 1) b n for all n sufficiently large. To use the Direct Comparison test to show divergence, we need to find a divergent series X n =1 c n for which ln n ( n 3 + 1) c n for all n sufficiently large. Notice that the numerator grows more slowly than n and the denominator grows with n 3 , so we suspect that the terms of the series will “behave like”
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: n n 3 (or 1 n 2 ) and that the series is likely convergent. To nd an upper bound b n , we bound the numerator above and the denominator below as follows: For all n 1, ln n < n and n 3 + 1 > n 3 from which it follows that 1 n 3 + 1 < 1 n 3 . Thus ln n n 3 + 1 < n n 3 = 1 n 2 . The series X n =1 1 n 2 is a convergent p series, therefore by the Direct Comparison Test, both series converge....
View Full Document

This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

Ask a homework question - tutors are online