MATH 2B Chapter Notes - Chapter 11.4: Monotone Convergence Theorem, User Friendly, Direct Comparison Test

• School: UC-Irvine
• Department: Mathematics
• Course: MATH 2B
• Professor: George Donaldson

Document Summary

The comparison test works, very simply, by comparing the series you wish to understand with one that you already understand. The proof relies on the monotone convergence theorem. Suppose, for ease of notation that 1 n < , and that 0 < an bn. Since an and bn are series of positive terms, they must either converge, or diverge to + . If sn = n i=1 ai and tn = n i=1 bi are the terms of the sequences of partial sums, then, being sums of nitely many positive terms, it is immediate that. 0 < sn tn < and that the sequences (sn) and (tn) are both increasing: suppose rst that bn = b converges. Then lim n sequence, bounded above by b. Therefore (sn) converges and so does an. tn = b. It follows that (sn) is an increasing: instead suppose that an = diverges. Then lim n whence bn diverges. sn = + .