Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
This follows from the de®nition of convergence. Then, with M n ju n …x†j for all x
in [a, b],
Hence
jS…x†s n …x† j ˆ
X 1
iˆN‡1
X 1
iˆN‡1
ju i …x† j 1), the series is uniformly and absolutely convergent for all x by
the M test.
(b) By the ratio test, the series converges in the interval 1 x 1 (or jxj 1).
For all x in jxj 1; x n =n 3=2
ˆ jxj n =n 3=2 1=n 3=2 . Choosing M n ˆ 1=n 3=2 ,
we see that P M n converges. So the given series converges uniformly for
jxj 1 by the M test.
522