Spring 2008, Math 152 honors
Exam 3 Overview
Chapter 10. (all but 10.8). You need to know and be able to apply the following:
1. definitions: a sequence, the limit of a sequence ( − δ form), the convergence /divergence of a sequence, increasing/deacreasing sequence, bounded sequence. (To test for
increase/decrease, it might help to find a derivative of f (x) such that f (n) = an .)
2. all limit laws and theorems about sequences (in particular, know and be able to apply
theorem stating that bounded monotonic sequence converges).
3. definitions: series, partial sums, convergent /divergent
P∞series, sum of a series. Be able
to
find
a
sum
of
a
telescoping
series
(such
as,
say,
1 sin(π/n) − sin(π/(n + 1)) or
P∞ 1/n
P∞
n
1/(n+1)
−5
or 0 ln n+1 , etc. You may need a partial fraction decomposition
1 5
P
1
to see that the series is telescoping, as for ∞
1 9n2 −1 )
4. (n-th term) test for divergence. Examples: Could you use this test to make a defiP n−1 2n
? Could you
nite conclusion about convergence/divergence of the series ∞
2
n
use
to make a definite conclusion about convergence/divergence of the series
P∞this test
√ 1 √ ?
2
n+1+ n
5. important series: geometric series, harmonic, p-series. Know when these series converge/diverge and how to justify your conclusion (what test is used to show convergence/divergence?). Be able to find the sum of a geometric series. What can be
P 5(−2)n
n+1
said about convergence of the series ∞
+ 4(−1)
and what is its sum, if it
2
3n+1
5n
converges?
6. tests for series with positive terms: integral test (with the remainder estimate), comparison test, limit comparison test, ratio test, root test. When use a test, especially
the integral test, make sure you checked that conditions for it are satisfied (f must
be positive decreasing function such that f (n) = an ). (try all examples in Stewart,
handout and Challenge 3)
7. definition of alternating
series, alternating series test, remainder estimate. Example:
P∞
2n
Does the series 2 (−1)n 1+n
satisfy the alternating series test? Estimate the error
2
made when the sum of the series is approximated by the 3rd partial sum.
8. absolute convergence. Relation between absolute convergence and convergence. For
a given series,
whetherPit converges absolutely, conditionally, or diverges.
P∞determine
2
n
n 2n
Examples: 2 (−1) (1 − n13 )n , ∞
2 (−1) 1+n2 .
9. power series. Finding the interval of convergence (2 steps: radius, and then the behavior of the series at end points of the found interval). Example: Find the radius an the
P
n
n
n 3 (2x−3)
interval of convergence: ∞
0 (−1) (2n)!(1+n2 )
10. expanding a function into a power series using ”tricks” such as sum of geometric series
1
1
5
(this is how you obtain the series for 1+x
, 1+x
2 , 2+3x , etc.;
operations on power series, including differentiation and integration. new series obtained by differentiation or integration of the old ones. (this is how you obtain the
series for ln (1 + x), arctan x, etc.)
11. Taylor and Maclaurin series. Taylor polynomial with the remainder estimate. Representation of functions by power series and applications of such representation. You
1
).
should be able to derive all representations on page 630 (ex , sin x, cos x, tan−1 x, 1±x
You should be able to derive Taylor/ Maclaurin series for other functions, too, using
the definition of the series or using the series for basic functions.
12. applications of Taylor series and Taylor polynomials: approximating functions, finding
limits, derivatives of a function at a point, estimating values of functions, estimating integrals, etc., including error estimate (using either alternating series remainder
estimation theorem or Taylor inequality).
Try yourself on answering questions below.
1. What does it mean that a sequence converges? Illustrate on the example: Show that
5n
= 5 by showing that for any , (try = 0.01 to get an idea), it is possible
limn→∞ n+1
5n
to find N such that | n+1
− 5| < 2. Is it true for a given sequence an that if |an | converges than an also converges? Is it true
for a given sequence an that if an converges than |an | also converges? If a statement is
true, justify (prove). If false, a counterexample is enough.
P
n
3. What is the definition of the convergent series? Use it to find the sum ∞
1 (ln( n+1 )
P
4. Is it true that if limn→∞ an =P
0 then the series ∞
1 an converges? Is it true that if
a
diverges?
Justify
your answer if true, or provide
limn→∞ an 6= 0 then the series ∞
n
1
a counterexample when false.
5. Define geometric series. Derive the formula for the sum of a geometric series (first for
the finite sum, then for infinite series). State the criteria for convergence of an infinite
geometric series. Use the formula to compute the sum of the given series if converges.
6. Define absolute convergence of a series. State the theorem about relation between
absolute convergence and convergence of a series. For a given series, determine the
type of convergence (absolute, conditional, or divergence).
P
P
7. TrueP
or false: if
cn diverges, then
|cn | also diverges.P(if the statement is false
and
|cn | converges, what conclusion can be made about
cn ? does this conclusion
agree with the assumption?)
P
8. What can be said about the convergence set of a power series cn (x + 3)n ? Could you
suggest cn so that the series converges only at one point? Could you suggest cn so that
the series converges everywhere? Could you suggest cn so that the series converges on
an interval (−a + R, a + R)?
9. What is Taylor polynomial (and what is the remainder); Taylor series; Maclauren
series? What is Taylor inequality? Why do we want Rn → 0 as n → ∞? What
condition guarantees that Rn → 0 as n → ∞?
10. Use Maclauren series to find f 10 (0) for f (x) = x2 sin x2 ; find limx→∞
P
2n+1
sum of the series 0→∞ (−1)n (π/2)
, etc.
2n+1
1−cos2x
;
1+2x−e2x
find the
11. Can the Maclauren series for arctan x (obtained by integrating the series expansion
1
at 0 for 1+x
2 ) be used to compute arctan π by substituting π into the series instead of
x? Why Yes or why No?
Suggested problems: Page 646, 1-62, also, Challenge 3 and Stewart handout problems. Take a look at past common exams when you think you are ready (not before!)
Good luck!