Sequences
1 hr 39 min 15 Examples
- Introduction to Sequences
- Write the first five terms of the sequence (Examples #1-3)
- Convergence: Limit of a Sequence Theorem
- Determine if the sequence converges by finding the limit of the sequence (Examples #4-6)
- Factorial notation and simplifying factorial expressions with examples
- Properties of limits of sequences
- Determine the convergence of the sequence (Examples #7-10)
- Understanding monotonic sequences (Examples #11-12)
- Understanding bounded sequences
- Monotonic Sequence Theorem for convergence (Examples #13-15)
Series
26 min 5 Examples
- Introduction to series and summation notation
- Summation properties
- Evaluate the finite series (Examples #1-3)
- Find the sum (Examples #4-5)
- Three important questions about convergence of an infinite series
Nth Term Test
34 min 11 Examples
- Overview of the nth term test (divergence test)
- Determine if the series diverges using the nth term test (Examples #1-3)
- Apply the nth term test to determine divergence (Examples #4-8)
- Apply the divergence test to the infinite series (Examples #9-11)
P-Series Test
29 min 9 Examples
- Overview of the p-series test
- Determine convergence using the p-series test (Examples #1-3)
- Use the p-series test to determine convergence (Examples #4-6)
- Apply the p-series test to find convergence or divergence (Examples #7-9)
Geometric Series
1 hr 3 min 9 Examples
- Overview of the geometric series
- Determine convergence; if convergent, find the sum (Example #1)
- Find the sum of the convergent geometric series (Examples #2-5)
- Determine if the series converges and find what it converges to (Examples #6-9)
Direct and Limit Comparison Tests
1 hr 42 min 12 Examples
- Overview of the direct comparison test
- Steps for using direct comparison
- Use direct comparison to determine whether the infinite series converges (Examples #1-2)
- Apply direct comparison to determine convergence (Examples #3-6)
- Overview of the limit comparison test
- Use both direct comparison and limit comparison to determine convergence (example)
- Use limit comparison to determine convergence (Examples #7-10)
- Apply limit comparison to determine convergence or divergence (Examples #11-12)
Integral Test
1 hr 10 min 6 Examples
- Overview of the integral test
- Apply the integral test to determine convergence (Example #1)
- Use the integral test to determine convergence or divergence (Examples #2-3)
- Use integration by parts and inverse trig integration to determine convergence (Examples #4-5)
- Use partial fractions and integration to determine convergence (Example #6)
Telescoping Series
50 min 6 Examples
- Overview of the telescoping series
- Use the telescoping series to find the sum of the series (Example #1)
- What is the sum of the convergent telescoping series? (Examples #2-3)
- If possible, find the sum of the infinite series (Examples #4-5)
- Determine the sum of the series using telescoping (Example #6)
Alternating Series Test
1 hr 2 min 11 Examples
- Overview of the alternating series
- Test for convergence using the alternating series test (Examples #1-2)
- Determine if the alternating series converges (Examples #3-6)
- Test for convergence by applying the alternating series test (Examples #7-8)
- Absolute convergence and conditional convergence for the alternating series
- Classify absolute or conditional convergence for the alternating series (Examples #9-11)
Ratio Test
1 hr 22 min 10 Examples
- Understanding the ratio test
- Use the ratio test to determine convergence or divergence (Examples #1-2)
- Determine convergence or divergence using the ratio test (Examples #3-5)
- Determine if the infinite series converges absolutely using the ratio test (Examples #6-8)
- Apply the ratio test and L’Hôpital’s rule to show absolute convergence (Example #9)
- When the ratio test is inconclusive (Example #10)
Root Test
41 min 7 Examples
- Overview of the root test
- Determine convergence or divergence using the root test (Examples #1-2)
- Use the root test to determine if the infinite series converges (Examples #3-4)
- Use the root test and the definition of e to determine if the series converges (Example #5)
- Determine if the infinite series converges using the root test (Example #6)
- Decide between the ratio test and the root test to determine convergence (Example #7)
Sequences and Series Review
1 hr 4 min 14 Examples
- Which of the following sequences converge? (Example #1)
- Find the formula for the nth term of the sequence (Example #2)
- Determine convergence for the infinite series (Examples #3-6)
- Does the infinite series converge? (Examples #7-9)
- Find the sum of the infinite series (Example #10)
- Which of the following series converge? (Examples #11-12)
- Find the sum of the series (Example #13)
- For what value of k will both series converge? (Example #14)
Radius and Interval of Convergence
2 hr 9 Examples
- Understanding power series
- Overview of radius and interval of convergence
- Determine the radius and interval of convergence (Example #1)
- Find the radius and interval of convergence for the power series (Example #2)
- Determine the radius and interval of convergence (Examples #3-4)
- For the given power series, find the radius of convergence (Examples #5-6)
- Find the convergence set for the power series (Example #7)
- Find the interval of convergence for the power series (Example #8)
- For the given power series, find the convergence set (Example #9)
Power Series
1 hr 10 min 9 Examples
- Basic form of a power series and power series representation
- Find the power series representation (Examples #1-4)
- Determine the power series representation for the function (Examples #5-7)
- Use partial fractions to find the power series representation (Example #8)
- Find the power series representation using partial fractions (Example #9)
Power Series Representation
1 hr 29 min 8 Examples
- Power series representation: differentiation and integration
- Find the power series by differentiating (Example #1)
- Use differentiation to find the power series (Example #2)
- Find the power series representation by using differentiation (Example #3)
- Find the power series by differentiating (Example #4)
- Find the power series by integrating (Example #5)
- Use integration to find the power series (Example #6)
- Find the power series representation by using integration (Example #7)
- Find the power series by integrating two functions (Example #8)
Taylor Series
2 hr 22 min 17 Examples
- What is a Taylor series? Taylor polynomial? Key observations
- Find the Taylor polynomial and series centered at a (Examples #1-2)
- Find the Taylor polynomial centered at a (Examples #3-4)
- Find a degree three Taylor polynomial for √x (Example #5)
- Six basic Maclaurin series expansion formulas
- Use the Maclaurin expansion to find the first five terms and the general term (Examples #6-7)
- Find the first five terms and the general term using the Maclaurin expansion formulas (Examples #8-9)
- Find the first five terms and the general term of the series (Example #10)
- Determine the first five nonzero terms using the Maclaurin expansions (Example #11)
- Use series to evaluate the limit (Examples #12-13)
- Use series to evaluate the limit (Example #14)
- Use series to approximate the definite integral to the indicated accuracy (Example #15)
- Approximate the definite integral using ten partial sums and round to the thousandths place (Example #16)
- Use series to approximate the integral accurate to four decimal places (Example #17)
Binomial Series
34 min 3 Examples
- Overview of the binomial theorem and the binomial series
- Using the binomial series instead of the binomial theorem
- Example #1: Write the first four terms of the binomial series
- Example #2: Write the first four terms of the binomial series
- Example #3: Write the first four terms of the binomial series
Alternating Series Error Bound
1 hr 2 min 7 Examples
- Overview of the alternating series remainder theorem
- Approximate the sum of the alternating series by its first six terms (Example #1)
- Use the alternating series estimation theorem to find the error using the 10th partial sum (Example #2)
- What is the error of the alternating series approximated by 15 terms? (Example #3)
- How many terms of the alternating series are needed to find the sum to the indicated accuracy? (Example #4)
- Find the number of terms of the alternating series needed to find the sum to the indicated accuracy (Example #5)
- Approximate the sum of the alternating series accurate to four decimal places (Example #6)
- Approximate the sum of the alternating series accurate to six decimal places (Example #7)
Lagrange Error Bound
2 hr 12 min 13 Examples
- What is error? Review of how to find alternating series error
- Overview of Taylor series remainder theorem, Taylor’s inequality, and Lagrange error bound
- Find the error bound for the function (Example #1)
- Find the upper bound for the error of the 5th degree polynomial of e (Example #2)
- Use the sixth order polynomial to approximate cos(2) and find the error bound (Example #3)
- What is the smallest value of k for the Lagrange error bound? (Examples #4-5)
- Approximate the function using the Taylor polynomial and estimate the error (Example #6)
- Find the error in estimating the function (Example #7)
- What is the error bound for the Taylor polynomial? (Example #8)
- Use the Lagrange error bound to show the approximation is accurate (Example #9)
- Estimate the error (Example #10)
- What is the error bound for the alternating series? (Example #11)
- Given the graph, find the Taylor polynomial, approximate, and show the error is accurate (Example #12a-c)
- Write a third degree Taylor polynomial for the function (Example #13a)
- Use integration to find the fourth degree polynomial (Example #13b)
- Find the radius of convergence for the function (Example #13c)
- Find the error using the sixth degree Taylor polynomial (Example #13d)
Chapter Test
3 hr 8 min 39 Examples
- 39 challenging practice problems
- Great for checking your knowledge
- Perfect for preparing for an in-class assessment