Sample Problems for Test II
NOTE: The test will contain problems that require straightforward computation, problems that require some
thought, as well as questions about basic definitions. The best strategy to prepare for the test is to make
sure that you (i) understand the basic concepts introduced in each section, (ii) know how to do (by yourself)
as many of the following as possible: the homework and quiz problems, the worked examples in the text, the
sugggested problems, and the following sample problems.
The test will cover sections 8.2 - 8.10 (excluding the root test in section 8.6) and 9.1 - 9.2. The key ideas
are the following.
Infinite Series
1. You should know the difference between a sequence and a series.
2. Given the nth term of a series, you should be able to write down the term for a particular value of
n. Given the first few terms, you should be able to identify the pattern and write down the formula
for the nth term.
3. You should understand what is meant by the sum of a series. Definitions of nth partial sum, conver-
gence and divergence.
4. You should know when a geometric series converges and what its sum is.
5. Convergence tests: (i) nth term test, (ii) integral test (and as a corollary, the p-series test), (iii) alter-
nating series test, (iv) ratio test. You should be able to estimate the error involved in approximating
an alternating infinite series by a finite sum.
6. Definitions of absolute and conditional convergence.
7. Check out pages 601 and 602 of the text about strategies for testing a series for convergence or
divergence.
Taylor Polynomials
1. Definition of a Taylor polynomial. You should know how to construct the Taylor polynomial Tn(x) of
a given degree n, for a given f(x),about a given center c.
2. You should know what the formula for the remainder Rnis, and how it can be used to estimate the
error of approximating fby Tn.
Power series
1. You should know how to test a power series for comvergence. Learn the definitions of, and the method
of finding, radius of convergence and interval of convergence.
2. Understand that within its interval of convergence, a power series represents a function, and that within
the radius of convergence the power series can be differentiated or integrated without change in radius
of convergence.
3. Binomial series.
4. Constructing new power series from old by adding, multiplying, integrating or differentiating known
series.
Conics and parametric curves
1. You should be able to derive the equation of a conic from given information.
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