Expect to be asked to state formal definitions for: (a)
limit as n --> infinity (a_n) = L; (b) limit as n -->
infinity (a_n) = + infinity; (c) limit as n --> infinity
(a_n) = - infinity
Expect to be asked to do proofs based on these formal
definitions
Be able to work with recursively defined sequences as well
as sequences defined by a "function rule" (see p617)
Be sure you understand the difference between a sequence
and a series (in particular, difference between sequence a_1,
a_2, a_3, ..., a_n ... and series a_1+a_2+a_3+...+a_n+...)
Know what it means for a sequence to converge
Know what is meant by the term sequence for a given series
and the partial-sum sequence for a given series, and know what
it means for a series to converge
Know how to find the partial-sum sequence when term
sequence is given (although, in general, it's usually difficult
to find a closed-form expression for the partial-sum sequence)
Know how to find the term sequence when the partial-sum
sequence is given
Know how to use the definition of series convergence to
determine whether given series converges (when it's feasible to
do so, which it often isn't)
Know how to find partial sums for and decide
convergence/divergence for telescoping series
Know the limit theorems for sequences, how to use them, and
how not to use them (or what they say and what they don't say)
Includes limit of a sum, product, etc (Theorem 2 on p622)
Includes Sandwich Theorem or Squeeze Theorem for Sequences
and using it to validate intuitive reasoning (see class notes
and Theorem 3 and Exple 2 p 623)
Includes understanding the relation between limit of a
sequence and limits of its subsequences (see class notes and pp
617-618)
Includes understanding the relation between limit as
x-->infinity of f(x) and limit as n-->infinity of f(n)
and why/how this allows us to use l'Hospital's rule on
sequences, when it's relevant (see class notes and pp 623-625)
Includes using continuity to justify changing limit of f(*)
to f(limit of *), what the text calls The Continuous Function
Theorem for Sequences (Theorem 4 p623)
From Limits That Arise Frequently (Table 8.1 on p625): know
the first five and how to find or verify them; know the last
one but, for now, you don't need to know how to verify it
Know Convergence Theorems for Series or Theorems on
Combining Series (Theorem 7 p636)
Know about reindexing series: how to do it; it does not
affect convergence/divergence; it does affect the sum of a
convergent series
Know about adding or deleting terms on a series: finitely
many new or different terms don't affect convergence/divergence
but do affect sum of convergent series
OK to use calculators but calculator evidence alone is not
enough when you're finding limits