Math Department Series &vector analysis Math205 Science collage
King Abdul Aziz University Dr. Najwa Joharji
Objectives of chapter 11 Infinite Sequences & Series
11.1 Sequences:
1. Define an infinite sequence.
2. Describe the sequence by writing rule that specify its terms.
3. Describe the sequence by listing its terms.
4. Show that a sequence is convergent.
5. Define the Sandwich Theorem for sequences.
6. Apply the continuous function theorem to show that a sequence is convergent.
7. Determine the convergence of a sequence using L’Hopital’s Rule.
8. Use Theorem 11.1.5 to find limits of sequences.
9. Define a nondecreasing sequence.
10. Investigate whether a sequence is increasing or decreasing
11. Use the Nondecreasing Sequence theorem to prove that the sequence converges.
11.2 Infinite Series:
1. Define an infinite series.
2. Define the partial sum sequence of a series.
3. Verify that a geometric series
is convergent or divergent.0 k
ark
4. Find the sum of the convergent geometric series
5. Express the repeating decimal as a ratio of two integers.
6. Find the sum of a telescoping series.
7. Use the Divergent test to show that a series is divergent.
8. Combine convergent series to make new convergent series.
9. Determine the convergence or divergence of the sum and subtraction of two series.
11.3 The IntegralTest:
1. Verify the conditions of the integral test.
2. Use the integral test to determine if a series converge or not.
3. Prove that the Harmonic series is divergent using the integral test.
4. Use the Integral Test to determine when the P-series is convergent or divergent.
11.4 Comparison Tests
:1. Verify the conditions of the Comparison Test.
2. Use the Comparison Test to determine if a series converges or diverges.
3. Verify the conditions of the Limit Comparison Test.
4. Investigate the convergence of a series using the Limit Comparison test.
Math Department Series &vector analysis Math205 Science collage
King Abdul Aziz University Dr. Najwa Joharji
11.5 The Ratio &Root Tests
:1. Use the Ratio Test to determine whether a series converges or not.
2. Investigate the convergence of a series using the Ratio Test.
3. Define the Root Test.
4. Use the Root Test to determine the convergence of a series.
11.6 Alternating Series, Absolute and Conditional Convergence
: 1. Define the Alternating Series.2. Verify the conditions of the Alternating Series test.
3. Use The Alternating Series test to determine the convergence of a series.
4. Define the Absolutely convergent series.
5. Define the Conditionally convergent series
6. Determine whether the series is absolutely convergent or conditionally convergent or Divergent.
11.7 Power Series
:1. Define a power series about x=a by
.
0 n
n
n x a
c
2. Define the radius of convergence and convergence interval.
3. Define the Geometric power series
and discuss where it converges.
0 n
xn
4. Test power series for convergence by Ratio Test.
5. Illustrate how we usually test a power series for convergence.
6. Apply Term by Term differentiation to find power series for f(x). 7. Apply Term by Term Integration to find power series for
f(x)dx.11.8 Taylor and Maclaurin Series:
1. Define the Taylor series of a function f at x=a
kk k
a k x
a
f
0 ) (
! )
( .
2. Define the Maclaurin series generated by f
kk k
k x
f0 ) (
! ) 0
( .
Math Department Series &vector analysis Math205 Science collage
King Abdul Aziz University Dr. Najwa Joharji
3. Find the Taylor series generated by f about x=a.
4. Define the Taylor polynomial pn(x) generated by a function f.
5. Find the Taylor polynomial pn(x) generated by a function f.
11.9 Convergence of Taylor Series; Error Estimation:
1. Estimate the sum of convergent Alternating series using the Alternating Series Estimation Theorem.
2. Find the error term using Taylor’s Formula f(x) pn(x)Rn(x) where
1 )
1 (
) )! (
1 (
) ) (
(
n n
n x a
n c x f
R for some c between a and x.
3. Estimate the remainder using the Reminder Estimation Theorem.
4. Use the Reminder Estimation Theorem to determine the accuracy of convergence.
5. Find a Taylor series by substitution.
6. Find a Taylor series by multiplication.
7. Decide how many terms to use to approximate a function to a given degree of accuracy.
8. Verify that Taylor series can be added, subtracted, and multiplied by constant and the result are once again Taylor series.
9. Define Euler’s identityei cos isin .
11.10 Applications of Power series:
1. Define the Binomial series for -1<x<1 by k
k
m x
k x
m
1
) 1
( .
2. Use Taylor series to express nonelementary integrals in terms of series.
3. Estimate a definite integral using estimation Theorems.
4. Evaluate limits using power series.
Math Department Series &vector analysis Math205 Science collage
King Abdul Aziz University Dr. Najwa Joharji
11.11 Fourier Series:
1. Define Fourier series Expansion for a function f(x)
1 1
0 cos sin
) (
k k k
k kx b kx
a a
x f
2. Find the Fourier series Expansion for a function f(x).
