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6.3 Taylor and maclaurin series  (Page 2/13)

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Uniqueness of taylor series

If a function f has a power series at a that converges to f on some open interval containing a , then that power series is the Taylor series for f at a .

The proof follows directly from [link] .

To determine if a Taylor series converges, we need to look at its sequence of partial sums. These partial sums are finite polynomials, known as Taylor polynomials    .

Visit the MacTutor History of Mathematics archive to read brief biographies of Brook Taylor and Colin Maclaurin and how they developed the concepts named after them.

Taylor polynomials

The n th partial sum of the Taylor series for a function f at a is known as the n th Taylor polynomial. For example, the 0th, 1st, 2nd, and 3rd partial sums of the Taylor series are given by

p 0 ( x ) = f ( a ) , p 1 ( x ) = f ( a ) + f ′ ( a ) ( x − a ) , p 2 ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 , p 3 ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ‴ ( a ) 3 ! ( x − a ) 3 ,

respectively. These partial sums are known as the 0th, 1st, 2nd, and 3rd Taylor polynomials of f at a , respectively. If x = a , then these polynomials are known as Maclaurin polynomials for f . We now provide a formal definition of Taylor and Maclaurin polynomials for a function f .

Definition

If f has n derivatives at x = a , then the n th Taylor polynomial for f at a is

p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ‴ ( a ) 3 ! ( x − a ) 3 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n .

The n th Taylor polynomial for f at 0 is known as the n th Maclaurin polynomial for f .

We now show how to use this definition to find several Taylor polynomials for f ( x ) = ln x at x = 1 .

Finding taylor polynomials

Find the Taylor polynomials p 0 , p 1 , p 2 and p 3 for f ( x ) = ln x at x = 1 . Use a graphing utility to compare the graph of f with the graphs of p 0 , p 1 , p 2 and p 3 .

To find these Taylor polynomials, we need to evaluate f and its first three derivatives at x = 1 .

f ( x ) = ln x f ( 1 ) = 0 f ′ ( x ) = 1 x f ′ ( 1 ) = 1 f ″ ( x ) = − 1 x 2 f ″ ( 1 ) = −1 f ‴ ( x ) = 2 x 3 f ‴ ( 1 ) = 2

Therefore,

p 0 ( x ) = f ( 1 ) = 0 , p 1 ( x ) = f ( 1 ) + f ′ ( 1 ) ( x − 1 ) = x − 1 , p 2 ( x ) = f ( 1 ) + f ′ ( 1 ) ( x − 1 ) + f ″ ( 1 ) 2 ( x − 1 ) 2 = ( x − 1 ) − 1 2 ( x − 1 ) 2 , p 3 ( x ) = f ( 1 ) + f ′ ( 1 ) ( x − 1 ) + f ″ ( 1 ) 2 ( x − 1 ) 2 + f ‴ ( 1 ) 3 ! ( x − 1 ) 3 = ( x − 1 ) − 1 2 ( x − 1 ) 2 + 1 3 ( x − 1 ) 3 .

The graphs of y = f ( x ) and the first three Taylor polynomials are shown in [link] .

This graph has four curves. The first is the function f(x)=ln(x). The second function is psub1(x)=x-1. The third is psub2(x)=(x-1)-1/2(x-1)^2. The fourth is psub3(x)=(x-1)-1/2(x-1)^2 +1/3(x-1)^3. The curves are very close around x = 1.
The function y = ln x and the Taylor polynomials p 0 , p 1 , p 2 and p 3 at x = 1 are plotted on this graph.
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Find the Taylor polynomials p 0 , p 1 , p 2 and p 3 for f ( x ) = 1 x 2 at x = 1 .

p 0 ( x ) = 1 ; p 1 ( x ) = 1 − 2 ( x − 1 ) ; p 2 ( x ) = 1 − 2 ( x − 1 ) + 3 ( x − 1 ) 2 ; p 3 ( x ) = 1 − 2 ( x − 1 ) + 3 ( x − 1 ) 2 − 4 ( x − 1 ) 3

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We now show how to find Maclaurin polynomials for e x , sin x , and cos x . As stated above, Maclaurin polynomials are Taylor polynomials centered at zero.

Finding maclaurin polynomials

For each of the following functions, find formulas for the Maclaurin polynomials p 0 , p 1 , p 2 and p 3 . Find a formula for the n th Maclaurin polynomial and write it using sigma notation. Use a graphing utilty to compare the graphs of p 0 , p 1 , p 2 and p 3 with f .

  1. f ( x ) = e x
  2. f ( x ) = sin x
  3. f ( x ) = cos x
  1. Since f ( x ) = e x , we know that f ( x ) = f ′ ( x ) = f ″ ( x ) = ⋯ = f ( n ) ( x ) = e x for all positive integers n . Therefore,
    f ( 0 ) = f ′ ( 0 ) = f ″ ( 0 ) = ⋯ = f ( n ) ( 0 ) = 1

    for all positive integers n . Therefore, we have
    p 0 ( x ) = f ( 0 ) = 1 , p 1 ( x ) = f ( 0 ) + f ′ ( 0 ) x = 1 + x , p 2 ( x ) = f ( 0 ) + f ′ ( 0 ) x + f ″ ( 0 ) 2 ! x 2 = 1 + x + 1 2 x 2 , p 3 ( x ) = f ( 0 ) + f ′ ( 0 ) x + f ″ ( 0 ) 2 x 2 + f ‴ ( 0 ) 3 ! x 3 = 1 + x + 1 2 x 2 + 1 3 ! x 3 , p n ( x ) = f ( 0 ) + f ′ ( 0 ) x + f ″ ( 0 ) 2 x 2 + f ‴ ( 0 ) 3 ! x 3 + ⋯ + f ( n ) ( 0 ) n ! x n = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ + x n n ! = ∑ k = 0 n x k k ! .

    The function and the first three Maclaurin polynomials are shown in [link] .
    This graph has four curves. The first is the function f(x)=e^x. The second function is psub0(x)=1. The third is psub1(x) which is an increasing line passing through y=1. The fourth function is psub3(x) which is a curve passing through y=1. The curves are very close around y= 1.
    The graph shows the function y = e x and the Maclaurin polynomials p 0 , p 1 , p 2 and p 3 .
  2. For f ( x ) = sin x , the values of the function and its first four derivatives at x = 0 are given as follows:
    f ( x ) = sin x f ( 0 ) = 0 f ′ ( x ) = cos x f ′ ( 0 ) = 1 f ″ ( x ) = − sin x f ″ ( 0 ) = 0 f ‴ ( x ) = − cos x f ‴ ( 0 ) = −1 f ( 4 ) ( x ) = sin x f ( 4 ) ( 0 ) = 0.

    Since the fourth derivative is sin x , the pattern repeats. That is, f ( 2 m ) ( 0 ) = 0 and f ( 2 m + 1 ) ( 0 ) = ( −1 ) m for m ≥ 0 . Thus, we have
    p 0 ( x ) = 0 , p 1 ( x ) = 0 + x = x , p 2 ( x ) = 0 + x + 0 = x , p 3 ( x ) = 0 + x + 0 − 1 3 ! x 3 = x − x 3 3 ! , p 4 ( x ) = 0 + x + 0 − 1 3 ! x 3 + 0 = x − x 3 3 ! , p 5 ( x ) = 0 + x + 0 − 1 3 ! x 3 + 0 + 1 5 ! x 5 = x − x 3 3 ! + x 5 5 ! ,

    and for m ≥ 0 ,
    p 2 m + 1 ( x ) = p 2 m + 2 ( x ) = x − x 3 3 ! + x 5 5 ! − ⋯ + ( −1 ) m x 2 m + 1 ( 2 m + 1 ) ! = ∑ k = 0 m ( −1 ) k x 2 k + 1 ( 2 k + 1 ) ! .

    Graphs of the function and its Maclaurin polynomials are shown in [link] .
    This graph has four curves. The first is the function f(x)=sin(x). The second function is psub1(x). The third is psub3(x). The fourth function is psub5(x). The curves are very close around x=0.
    The graph shows the function y = sin x and the Maclaurin polynomials p 1 , p 3 and p 5 .
  3. For f ( x ) = cos x , the values of the function and its first four derivatives at x = 0 are given as follows:
    f ( x ) = cos x f ( 0 ) = 1 f ′ ( x ) = − sin x f ′ ( 0 ) = 0 f ″ ( x ) = − cos x f ″ ( 0 ) = −1 f ‴ ( x ) = sin x f ‴ ( 0 ) = 0 f ( 4 ) ( x ) = cos x f ( 4 ) ( 0 ) = 1.

    Since the fourth derivative is sin x , the pattern repeats. In other words, f ( 2 m ) ( 0 ) = ( −1 ) m and f ( 2 m + 1 ) = 0 for m ≥ 0 . Therefore,
    p 0 ( x ) = 1 , p 1 ( x ) = 1 + 0 = 1 , p 2 ( x ) = 1 + 0 − 1 2 ! x 2 = 1 − x 2 2 ! , p 3 ( x ) = 1 + 0 − 1 2 ! x 2 + 0 = 1 − x 2 2 ! , p 4 ( x ) = 1 + 0 − 1 2 ! x 2 + 0 + 1 4 ! x 4 = 1 − x 2 2 ! + x 4 4 ! , p 5 ( x ) = 1 + 0 − 1 2 ! x 2 + 0 + 1 4 ! x 4 + 0 = 1 − x 2 2 ! + x 4 4 ! ,

    and for n ≥ 0 ,
    p 2 m ( x ) = p 2 m + 1 ( x ) = 1 − x 2 2 ! + x 4 4 ! − ⋯ + ( −1 ) m x 2 m ( 2 m ) ! = ∑ k = 0 m ( −1 ) k x 2 k ( 2 k ) ! .

    Graphs of the function and the Maclaurin polynomials appear in [link] .
    This graph has four curves. The first is the function f(x)=cos(x). The second function is psub0(x). The third is psub2(x). The fourth function is psub4(x). The curves are very close around y=1
    The function y = cos x and the Maclaurin polynomials p 0 , p 2 and p 4 are plotted on this graph.
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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