Information on Math 5710. Spring 2009.
Synopsis of topics on convergence of infinite series Sect.34, 35, 36
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Definition of a sum of an infinite series ( Definition 34.1). Properties of convergent
series (Theorem 34.1). Convergence of an infinite series with positive terms (Theorem
36.4). Cauchy criteria for convergence of series (Theorem 34.5).
Definition 34.6 of absolutely convergent and conditionally convergent series. Tests for
absolute convergence: Comparison test (Theorem 35.1), Limit comparison test (Theorem
35.2), Root test (Theorem 35.3) and Ratio test (Theorem 35.6), Integral test (Theorem
35.12).
Conditional convergence of series: Abel lemma (Theorem 36.1), Dirichlet test (The-
orem 36.2). Alternating series : Definition 36.6 and Test (Theorem 36.7).
Homework problems:
Section 34: B, F, G, H, O. Section 35 : A (b)-(g), B, C (b)-(e)
Quiz problems - examples.
1. Establish the divergence or convergence of the following series
(a) xn=1
n+ 3)(n+ 2)2
(b) xn=n!/nn
(c) xn= (−1)nn/(n+ 1)
(d) xn= 2ne−n
(e) xn=n!e−n2
2. Find all values of pfor which the following series converges absolutely and
conditionally
∞
X
n=1
(−1)n
n+np
1