MAT 21C Lecture Notes - Lecture 8: Absolute Convergence, Ratio Test, Commutator Subgroup
52
MAT 21C Full Course Notes
Verified Note
52 documents
Document Summary
Announcements: ec1 due oct 11, hw3 due oct 12, midterm 1 oct 14. A power series about x = a is a series of the form cn(x a)n = c0 + c1(x a) + c2(x a)2 + + cn(x a)n + . Xn=0 in which the center a and the coe cients c0, c1, are constants. Theorem 18. (the convergence theorem for power series) if the power series anxn = a0 + a1x + a2x2 + . Xn=0 converges at x = c 6= 0, then it converges absolutely for all x wtih |x| < |c|. If the series diverges at x = d, then it diverges for all x with |x| > |d|. Example: use the ratio test (or root test) to nd the interval where the series converges absolutely. (2x)n. |(2x)| = |2x| converges when < 1 or .
