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
(17.317.5), 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
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(19.5).
Problems: prove that a given function is or is not uniformly
continuous (19.119.3 and 19.519.6).
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 (20.720.9), 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).
2
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 Spring '08
 Staff
 Calculus, Continuous function, Metric space, Cauchy

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