Here is a list of topics to be covered in Midterm II. 
You are responsible for all material from the lectures, 
homework, textbook sections and the posted notes.

SERIES:  sections 14, 15, posted notes on series
 
-- convergence of series (know definitions and basic examples, such 
as geometric series, harmonic series, etc, with proofs of 
convergence/divergence; be able to work with partial sums) 

-- divergence test: a convergent series must have a_n --> 0 
(know proof, understand why converse is not true)

-- comparison test (for series with positive terms only, see notes), 
versions for convergence and divergence. Know and understand proofs, 
be able to use the test for a given series to detect convergence/divergence

-- ratio test (for series with positive terms only and for the case when lim
of the ratio exists, as in the posted notes) -- know proof, 
be able to use the test for specific series. Know what happens when the limit of the ratio is 1. 

-- Integral test as described in section 15. Know proof of Thm 15.1

-- alternating series test (know statement and proof of thm 15.3, be able 
to apply the test in examples)  

***** material not covered in class is not on the test - in particular, the harder version of the ratio test discussed in the book is not included ***  

Extra practice for series: 14.1, 14.2, 14.3, 14.5, 14.6, 14.9, 14.11, 14.14, 
Example 9 p.103,  15.1, 15.2, 15.3, 15.4, 15.5



CONTUNUOUS FUNCTIONS: sections 17, 18

-- Definitions of continuity (via sequences and the epsilon-delta definition);
be able to state both, use them to prove/disprove from scratch in simple examples

-- Equivalence of the two definitions (Theorem 17.2) - know proof

-- Operations with continuous functions (theorems 17.3, 17.4, 17.5) - 
know proofs, be able to use these theorems to prove continuity of given functions

-- Theorem: a continuous function on [a,b] is bounded and attains 
max and min. Know proofs, know what happens when function is discontinous 
or the interval is not closed

-- Intermediate Value theorem: know statement, proof, be able to use it

**** theorems 18.4, 18.5, 18.6 are not on the test ****
 
Extra practice: Examples in section 17, questions 17.3, 17.4, 17.5, 17.6, 17.7, 17.9, 17.10, 18.1, 18.2, 18.5, 18.7, 18.8, homeworks  

LIMITS OF FUNCTIONS: section 20

-- Definition of the limit of function in terms of sequences and in terms of epsilon-delta
(you should be able to give each variant of the definition, for finite and infinite limits, 
at a point or at infinity, including one-sided limits)

-- Prove equivalence of the two definitions (Theorem 20.6 and other variants as above, see p.160)

-- Computing limits or showing that they don't exist directly from definitions (simple examples, 
all variants)

-- Limit of a function continuous at a point; uniqueness of limit (Remarks 20.2 on p.153, know proof)

-- Limit laws for functions (Theorem 20.4); you may be asked to prove a specific variant. Undertstand what happens when limits are infinite

-- Existence of two-sided limits via one-sided limits, Theorem 20.10 (know proof)


Extra practice: 20.1 - 20.10, 20.14, 20.16, 20.17, 20.19, 20.20, homework. Design your own questions 
like Q3 HW10.