Bruce Mayer, PE Licensed Electrical & Mechanical Engineer - PowerPoint Presentation

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Bruce Mayer, PELicensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

Chabot Mathematics

§10.3 Series:Power & Taylor

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Review §

Any QUESTIONS About§10.2 Convergence TestsAny QUESTIONS About HomeWork§10.2 → HW-18

10.2

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§10.3 Learning Goals

Find the radius and interval of convergence for a power seriesStudy term-by-term differentiation and integration of power seriesExplore Taylor series representation of functions

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Power Series

General Power Series:A form of a GENERALIZED POLYNOMIALPower Series Convergence BehaviorExclusively ONE of the following holds TrueConverges ONLY for x = 0 (Trival Case)Converges for ALL x Has a Finite “Radius of Convergence”, R

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Radius of Convergence

For the General Power SeriesUnless a power series converges at any real number, a number R > 0 exists such that the series CONverges absolutely for each x such that | x | < R and DIverges for any other xThus the “Interval of Convergence”

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Example  Radius of Conv.

Find R for the Series:Radius of ConvergenceInterval of ConvergenceSOLUTIONUse the Ratio Test

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Example  Radius of Conv.

Continue with Limit Evaluation:Thus R = 4The Interval of ConvergenceThus This SeriesConverges

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Functions as Power Series

Many Functions can be represented as Infinitely Long PolyNomialsConsider this Function and DomainRecall one of The Geometric SeriesThus

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Example  Fcn by Pwr Series

Write as a Power Series →Also Find the Radius of ConvergenceSOLUTION:Start with the GeoMetric SeriesFirst Cast the Fcn into the Form

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Example  Fcn by Pwr Series

Using Algebraic Processes on the FcnThus by the Geometric SeriesThen the Function by Power Series

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Example  Fcn by Pwr Series

Now find the Radius of Convergence by the Ratio Test

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Example  Fcn by Pwr Series

Thus for ConvergenceSo the Interval of Convergence:And also the Radius of Convergence

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Pwr Series Derivatives & Integrals

Consider a Convergent Power SeriesAnd an Associated FunctionIf f(x) is differentiable over −R<x<R, then

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Pwr Series Derivatives & Integrals

If f(x) is Integrable over −R<x<R, then

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Pwr Series Derivatives & Integrals

Thus the Derivative of a Power-Series FunctionThus the AntiDerivative of a Power-Series Function

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Example  Find Fcn by Integ

Find a Power Series Equivalent forSOLUTION:First take:Recognizefrom Before

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Example  Find Fcn by Integ

Recover the Original Fcn by taking the AntiDerivative of the Just Determined Derivative

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Example  Find Fcn by Integ

ThenTo Find C use the original FunctionUse f(0) = 0 in Power Series fcnThen the Final Power Series Fcn

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Taylor Series

Consider some general Function, f(x), that might be Represented by a Power SeriesThus need to find CoEfficients, an, such that the Power Series Converges to f(x) over some interval. Stated Mathematically Need an so that:

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Taylor Series

If x = 0 and if f(0) is KNOWN then a0 done, 1→∞ to go….Next Differentiate Term-by-TermNow if the First Derivative (the Slope) is KNOWN when x = 0, then

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Taylor Series

Again Differentiate Term-by-TermNow if the 2nd Derivative (the Curvature) is KNOWN when x = 0, then

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Taylor Series

Another DifferentiationAgain if the 3rd Derivative is KNOWN at x = 0 Recognizing the Pattern:

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Taylor Series

Thus to Construct a Taylor (Power) Series about an interval “Centered” at x = 0 for the Function f(x)Find the Values of ALL the Derivatives of f(x) when f(x) = 0Calculate the Values of the Taylor Series CoEfficients byFinally Construct the Power Series from the CoEfficients

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Example  Taylor Series for ln(e+x)

Calculate the DerivativesFind the Values of the Derivatives at 0

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Example  Taylor Series for ln(e+x)

GenerallyThen the CoEfficientsThe 1st four CoEfficients

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Example  Taylor Series for ln(e+x)

Then the Taylor Series

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Taylor Series at x ≠ 0

The Taylor Series “Expansion” can Occur at “Center” Values other than 0Consider a function stated in a series centered at b, that is:Now the the Radius of Convergence for the function is the SAME as before:

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Taylor Series at x ≠ 0

To find the CoEfficients need (x−b) = 0 which requires x = b, Then the CoEfficient ExpressionThe expansion about non-zero centers is useful for functions (or the derivatives) that are NOT DEFINED when x=0For Example ln(x) can NOT be expanded about zero, but it can be about, say, 2

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Example  Expand x½ about 4

Expand about b = 4:The 1st four Taylor CoEfficients

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Example  Expand x½ about 4

SOLUTION:Use the CoEfficients to Construct the Taylor Series centered at b = 4

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Example  Expand x½ about 4

Use the Taylor Series centered at b = 4 to Find the Square Root of 3

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WhiteBoard PPT Work

Problems From §10.3P39 → expand aboutb = 1 the Function

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All Done for Today

Brook

Taylor

(1685-1731)

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Bruce Mayer, PELicensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

Chabot Mathematics

Appendix

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Do On

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Borad

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P10.3-39 Taylor Series

Da1 := diff(ln(x)/x, x)Db2 := diff(Da1, x)Dc3 := diff(Db2, x)Dd4 := diff(Dc3, x)

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P10.3-39 Taylor Series

ln(x)/x, xf0 := taylor(ln(x)/x, x = 1, 0)f1 := taylor(ln(x)/x, x = 1, 1)f2 := taylor(ln(x)/x, x = 1, 2)

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P10.3-39 Taylor Series

f3 := taylor(ln(x)/x, x = 1, 3)f4 := taylor(ln(x)/x, x = 1, 4)d6 := diff(ln(x)/x, x $ 5)

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P10.3-39 Taylor Series

plot(f0, f1, f2, f3, f4, f5, x =0.5..3, GridVisible = TRUE,LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24],TicksLabelFont=["sans-serif", 16])

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