MA 126
Review for Exam 3
April 2015
Here are some things you should know for the exam. You are also expected to know relevant precalculus and
Calculus I material. For example, you must be able to compute limits of functions to use theorem 4 in 9.1
and to apply divergence, root and ratio and root tests. Review this material if necessary.
The exam is on Tuesday, April 14. Material covered: All of Chapter 9, except the Series Multiplication
Theorem (p. 534) and the Binomial Series (p. 551).
Sequences: Be able to apply Theorems 1–4 in section 9.1, together with basic limits from Theorem 5, to
P∞
find limits of sequences. Do not confuse sequences with series, or limn→∞ an with n=0 an .
Infinite series: Be prepared to state the definition of convergence for infinite series. Be able to write the
nth partial sum of a series. For telescoping series, you should be able to find a simple formula for sn and
find its limit. Recognize geometric series, know when they converge and the formula for the sum. Be able to
apply the sum, difference and constant multiple rules. Know what the nth-Term Test for Divergence says,
P
P
and what it doesn’t say: If an 6→ 0 then
an diverges, but if an → 0 the series
an may still diverge!
Understand and be able to correctly apply the theorems for convergence of positive series: Integral Test
(check that f is positive and decreasing), Limit Test and Limit Comparison Test, Ratio Test and Root Test.
Know when a p-series converges.
Know the definitions of absolute and conditional convergence. Know how and when to apply the Alternating
Series Test, and the Alternating Series Estimation Theorem. By working many exercises and thinking about
the techniques used to prove convergence, you should learn how to choose an effective test for a given series.
Power series and Taylor series: Know what a power series is, and how to find the radius of convergence
and interval of convergence. (Use the Ratio Test or Root Test to find the radius of convergence R, and
if 0 < R < ∞, use another test to check for convergence at the endpoints x − a = R and x − a = −R.)
Remember that |x − a| < R means a − R < x < a + R. Know how to use substitution and term-by-term
differentiation and integration to find power series representations of functions.
Be able to write the general formulas for the Taylor series and Taylor polynomials generated by f at x = a
(for a = 0 the Taylor series is also called the Maclaurin series.) Be able to find the Taylor polynomials from
the definition, and the Taylor series for simple examples where f (n) (a) has a simple pattern. Be able to state
Taylor’s formula for the remainder term Rn (x) and apply the Remainder Estimation Theorem.
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Know the Maclaurin series for ex , cos x, sin x and 1−x
, and where they converge. Be able to find other Taylor
series from these or from given series using substitution, term-by-term differentiation and integration. (If
you know the power series of sin x or cos x, you can get the other this way.) Understand how power series
are applied to obtain numerical estimates of functions and their integrals.