Overview
We will now extend the process that lead to Taylor polynomials, and discover Taylor series for many functions: Infinite series (involving a variable \(x\)) that give us new formulas for old functions.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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State the general formula for the \(n\)th order Taylor polynomial centered at \(x = a\), \(P_{n}(x)\).
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Describe some key features and purposes of a Taylor polynomial in words and in symbols. (For example: What is special about \(P_{n}'(a)\)? \(P_{n}''(a)\), etc.?)
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Given a function \(f(x)\) and a center \(x = a\), write a formula for the \(n\)th degree Taylor polynomial \(P_{n}(x)\) centered at \(x = a\). This includes correctly evaluating the derivatives \(f^{(k)}(a)\) and writing all other parts of the formula correctly.
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Be able to write a formula for the Taylor series for \(f\) centered at \(x = a\), using both summation notation and by writing a formula directly.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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State the meaning of the radius and interval of convergence for a series.
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Determine the interval and radius of convergence of a Taylor series. In particular, test the endpoints correctly.
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Fluently move between Taylor series and the functions they represent. In particular, evaluate infinite series by recognizing their Taylor series evaluated at a point.
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Calculate Taylor series for new functions by applying algebra and calculus to existing Taylor series.
To prepare for class
Use these resources to become proficient with the basic objectives (see above) before class.
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Read the rest of Section 8.5 in Active Calculus, focussing on Taylor series. Only skim the last subsection about the Error Approximations (the “Lagrange error bound”).
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(optionally) Review the videos from last class if necessary.
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Watch the following new videos:
(Finding the radius and interval of convergence)
After class
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Watch the end (starting at 17:13) of the following video by Grant Sanderson (“3Blue1Brown”):
