Using Mathematica to study series (part II)

Transcription

1 Using Mathematica to study series (part II) Truncation errors Example of exponential; first consider x>0 Remainder if use only first 3 terms in power series for exponential: PlotExpx1xx ^,x, 0,, PlotStyleThick, PlotLabel"Error in truncated Exponential ", AxesLabel"x", "" Error in truncated Exponential x Compare to bound discussed in class (plotted in red): PlotExpx1xx ^, x ^ 3 Expx6, x, 0,, PlotStyleBlue, Thick,Red, Thick, PlotLabel"Error in truncated Exponential vs bound", AxesLabel"x", "" Error in truncated Exponential vs bound x This plot shows that the bound works, although it becomes a poorer bound as x increases.

2 Series.nb PlotExpx1xx ^ x ^ 3 Expx6, x, 0.001,, PlotRange0, 1, PlotStyleRed, Thick, PlotLabel"Ratio of Error in truncated Exponential to bound", AxesLabel"x", "" 1.0 Ratio of Error in truncated Exponential to bound x PlotExpx, 1 xx ^,1xx ^ Expx x ^ 3 6,1xx ^ Expx x ^ 3 6, x, 0,, PlotStyleBlue, Thick,Red, Thick, None, None, Filling34, FillingStyleDirectiveOpacity.5, Brown, PlotLabel"Exponential vs Truncated Exponential with bound", AxesLabel"x", "" Here we can see how the region including the bound grows. Exponential for x<0 Compare error to general bound (which is the same as the alternating series bound in this case):

3 Series.nb 3 PlotExpx1xx ^, x ^ 3 6,x,, 0, PlotStyleBlue, Thick,Red, Thick,Green, Thick, PlotLabel"Error in truncated Exponential vs bounds ", AxesLabel"x", "" Error in truncated Exponential vs bounds x Again the bound works, but becomes increasingly poor as x increases. PlotExpx1xx ^ x ^ 3 6,x,,0.001, PlotRange0, 1, PlotStyleRed, Thick,Green, Thick, PlotLabel"Ratio of Error in truncated Exponential to bound", AxesLabel"x", "" Ratio of Error in truncated Exponential to bound x

4 4 Series.nb PlotExpx, 1 xx ^,1xx ^ x ^ 3 6,1xx ^ x ^ 3 6, x,,0.001, PlotStyleBlue, Thick,Red, Thick, None, None, Filling34, FillingStyleDirectiveOpacity.5, Brown, PlotLabel"Exponential vs Truncated Exponential with bound", AxesLabel"x", "" Example of PolyLog Here s a sum that Mathematica knows as to be a special function: Sumx ^ n n ^ 3,n, 1, Infinity PolyLog3, x Interval of convergence is 1 x 1, although function can be extended to more negative x as shown in the plot. The extension ("analytic continuation") to more positive x yields a complex function. PlotPolyLog3, x,x,10, 1, PlotStyleThick Converges at x=1 NPolyLog3, but beyond get complex result from analytic continuation! This is an example of the lack of convergence not being due to the function tending to infinity, but rather because of the presence of a different type of singularity, namely a "branch cut". We ll study these when we talk about complex functions.

5 Series.nb 5 PolyLog3, How well does the first three terms represent the function? Call the sum of the first three terms in Taylor series f3[x]: f3x_ : xx ^ 8 x ^ 3 7 To the eye, the first three terms give a good represenation in the range of convergence: PlotPolyLog3, x, f3x,x,1, 1, PlotStyleThick Once one goes further out, the difference becomes much larger (but note this is outside the range where the bound is derived, which is the range of convergence) PlotPolyLog3, x, f3x,x,5, 1, PlotStyleThick Here s the error. According to the results discussed in class, this is bounded by x^4/64 for 1 x 0 and x^4/(64(1 x)) for 0 x 1

6 6 Series.nb PlotPolyLog3, xf3x,x,1, 1, PlotStyleThick Check bound for negative x by looking at the ratio of the error to the bound It works for negative x (ratio is less than 1), but not for positive x. negboundx_ x ^ 4 64; PlotPolyLog3, xf3xnegboundx,x,1, 1, PlotStyleThick Similarly, bound for positive x works too (though not for negative x). posboundx_ x ^ 41x 64; PlotPolyLog3, xf3xposboundx,x,1, 1, PlotStyleThick

7 Series.nb 7 PlotPolyLog3, x, f3x, HeavisideThetax posboundxf3xheavisidethetax negboundx, HeavisideThetax posboundxf3xheavisidethetax negboundx, x,1, 1, PlotStyleBlue, Thick,Red, Thick, None, None, Filling34, FillingStyleDirectiveOpacity.5, Brown, PlotLabel"Polylog vs Truncated Polylog with bound", AxesLabel"x", "" Numerical example discussed in class: At x=1/, difference is indeed smaller than bound PolyLog3, 0.5, f30.5, PolyLog3, 0.5f30.5, posbound , , , At x= 1/, works also: PolyLog3,0.5, f30.5, PolyLog3,0.5f30.5, negbound , , , Developing and Multiplying Series Mathematica will develop power series (the second argument in the second parenthesis being the value of x about which the Taylor series is being expanded). SeriesCosx,x, 0, 0 1 x x4 x6 x8 x10 x1 x x x18 x0 Ox Here is one way to display only numerical values (using // N after the command). SeriesCosx,x, 0, 0 N 1.0.5x x x x x x x x x x0. 0 Ox0. 1 If you want to pick out the coefficient of one term in the series, here s a way to do so:

8 8 Series.nb CoefficientSeriesCosx,x, 0, 0, x ^ Here s a slightly different way: CoefficientSeriesCosx,x, 0, 0, x, One can expand about any value of x. Here is the expansion about 3 Pi/, so that one gets the Sin function coefficients: SeriesCosx,x, 3 Pi, 6 x 3Π 1 6 3Π x x 3Π 5 Ox 3Π 7 Here is an example of multiplying series (done in class by hand), and it is much easier using Mathematica if one wants high order terms SeriesExpx Cosx,x, 0, 10 1x x3 Dividing series: 3 x4 6 x5 x7 x8 x9 Ox SeriesSinhxCosx,x, 0, 10 x x3 3 3 x x x9 340 Ox11 You can use the function Normal[ ] to have Mathematica to return the function at that order without the Ox n notation, leaving a function that you can manipulate. NormalSeriesSinx,x, 0, 7 x x3 6 x5 x One caveat to this is you will need to use Evaluate[ ] to make sure Mathematica does the Taylor expansion before plugging in a value of x gx_ : NormalSeriesSinx,x, 0, 5 g Sin This doesnt work because Mathematica is trying to do an expansion in a variable but you have given it a number. gx_ : EvaluateNormalSeriesSinx,x, 0, 5 g 14 15

9 Series.nb 9 This does work. Here is an example of how this can be used show how the Taylor series gets closer to the actual function as you go higher order. ManipulatePlotEvaluateSinx, NormalSeriesSinx,x, 0, n, x,30, 30, PlotRange,, PlotStyleRed, Thick,Blue, Thick, PlotLabel"Sin vs Truncated Sin at order n",n, 1, 50, 1 n Sin vs Truncated Sin at order n Essential Singularity Here is what Mathematica does with an essential singularity, for which the Taylor series has vanishing coefficents. It gives back the function without bothering to give the Taylor expansion. SeriesExp1x ^,x, 0, 6 1 x Here s what the function looks like:

10 10 Series.nb PlotExp1x ^,x,5, 5, PlotStyle Thick