7 and 118 sample problems 3 series definitions

of subsequential limits (11.7 and 11.8). Sample problems:
3. Series. Definitions: convergent series, absolute convergence, Cauchy
property for series. Theorems: sum of a geometric series, absolute
convergence implies convergence, comparison test, ratio test, root test,
integral test, alternating series test. Sample problems: basically you
need to be able to use all of these tests and the two standard series
(
∑
1 /n s and
∑
1 /a n ) to prove that a given series converges or diverges.
There’s a large number of problems in the book.
4. Continuous functions and their properties. Definitions: continuous
function. Theorems: ε - δ definition of continuity (17.2), sums, prod-
ucts, ratios and compositions of continuous functions are continuous
(17.3-17.5), a continuous function on a closed interval is bounded and
has extrema (18.1), intermediate value theorem (18.2).
5. Uniform continuity. Definitions: uniformly continuous function. Theo-
rems: a continuous function on a closed interval is uniformly continuous
(19.2), Cauchy sequences (19.3), uniformly continuous functions extend
1

(19.5). Problems: prove that a given function is or is not uniformly
continuous (19.1-19.3 and 19.5-19.6).
6. Limits. Definitions: limit of a function along an arbitrary approach
set S (20.1) and its various specializations (20.2, 20.3). Theorems:
limit theorems (20.4 and 20.5) ε - δ definition of a limit (20.6), and
the various specializations (20.7-20.9), left and right limits versus limit
(20.10) Problems: you need to be able to write down the two different
definitions of a limit (using subsequences and ε - δ ) for any of the 15
different possibilities (limit at a finite point, left and right limits, limits
at infinity, and infinite limits). Discussion 20.9 is very useful here.
Also, you need to be able to prove that a given function has a given
limit at a point (from the definitions directly or using any of the limit
theorems).
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