We can use limits to formally define sums of infinitely many values. Taylor Series are infinite sums defined in terms of polynomials, and provide an alternate means of describing standard functions.
» Session 94: Infinite Series
» Session 95: Series Comparison
» Session 96: Stacking Blocks
» Session 97: Power Series
» Session 98: Taylor’s Series
» Session 99: Taylor’s Series, Continued
» Session 100: Operations on Power Series
» Session 101: Conclusion
Power series are like infinite polynomials. Can they be added? Multiplied? Integrated? If so, we can use power series to approximate functions like the error function.
Clip 1: Power Series Multiplication
Clip 2: Derivative of a Power Series
Clip 3: Integral of a Power Series
Clip 4: Substitution of Power Series
Summing Infinite Series by Comparing to Taylor Series
Clip 1: Power Series Expansion of the Error Function
The problems in this session bring together ideas from throughout the course. The session ends with a farewell and an invitation.
Hyperbolic Sine
In this session we move from finding the area of an infinitely long region to finding the sum of infinitely many terms; Riemann sums form a bridge between these two topics. The most important infinite series is the geometric series, whose value is surprisingly easy to describe.
Clip 1: Introduction to Infinite Series
Summing the Geometric Series
In this session we use Riemann sums to compare series to indefinite integrals. This allows us to determine whether the series converge and may help determine the value of a convergent series.
Clip 1: Comparison of the Harmonic Series
Clip 3: Examples of Comparison
Using the Ratio Test
In this session we apply infinite series to a mathematical puzzle. Professor Jerison stacks identical blocks so that each block extends beyond the one below it. Must some part of the top block always be above the bottom block, or can the stack extend further than the length of the bottom block?
Power series are like infinite polynomials. We can use what we know about infinite series to determine for what values of x a power series converges.
Clip 1: Introduction to Power Series
Finding the Radius of Convergence
Earlier we studied linear and quadratic approximations. If we continue to improve our approximations by using the third, fourth, fifth, … derivatives the result is a power series. These power series are called Taylor’s series. This session gives a formula describing the terms of a Taylor’s series and a few examples of its application.
Clip 1: Introduction to Taylor’s Series
Clip 3: Taylor’s Formula Examples
Proving That the Taylor Series of a Function is Equal to the Function
In this session we discuss convergence of an alternating Taylor’s series and calculate a few more examples.
Clip 1: Review of Taylor’s Series
Clip 2: Taylor’s Series of 1/(1 + x)
Clip 3: Taylor’s Series of sin(x)
