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6.4 Working with taylor series  (Page 2/11)

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∑ n = 0 ∞ ( r n ) x n = 1 + r x + r ( r − 1 ) 2 ! x 2 + ⋯ + r ( r − 1 ) ⋯ ( r − n + 1 ) n ! x n + ⋯ .

We now need to determine the interval of convergence for the binomial series [link] . We apply the ratio test. Consequently, we consider

| a n + 1 | | a n | = | r ( r − 1 ) ( r − 2 ) ⋯ ( r − n ) | x | | n + 1 ( n + 1 ) ! · n | r ( r − 1 ) ( r − 2 ) ⋯ ( r − n + 1 ) | | x | n = | r − n | | x | | n + 1 | .

Since

lim n → ∞ | a n + 1 | | a n | = | x | < 1

if and only if | x | < 1 , we conclude that the interval of convergence for the binomial series is ( −1 , 1 ) . The behavior at the endpoints depends on r . It can be shown that for r ≥ 0 the series converges at both endpoints; for −1 < r < 0 , the series converges at x = 1 and diverges at x = −1 ; and for r < −1 , the series diverges at both endpoints. The binomial series does converge to ( 1 + x ) r in ( −1 , 1 ) for all real numbers r , but proving this fact by showing that the remainder R n ( x ) → 0 is difficult.

Definition

For any real number r , the Maclaurin series for f ( x ) = ( 1 + x ) r is the binomial series. It converges to f for | x | < 1 , and we write

( 1 + x ) r = ∑ n = 0 ∞ ( r n ) x n = 1 + r x + r ( r − 1 ) 2 ! x 2 + ⋯ + r ( r − 1 ) ⋯ ( r − n + 1 ) n ! x n + ⋯

for | x | < 1 .

We can use this definition to find the binomial series for f ( x ) = 1 + x and use the series to approximate 1.5 .

Finding binomial series

  1. Find the binomial series for f ( x ) = 1 + x .
  2. Use the third-order Maclaurin polynomial p 3 ( x ) to estimate 1.5 . Use Taylor’s theorem to bound the error. Use a graphing utility to compare the graphs of f and p 3 .
  1. Here r = 1 2 . Using the definition for the binomial series, we obtain
    1 + x = 1 + 1 2 x + ( 1 / 2 ) ( − 1 / 2 ) 2 ! x 2 + ( 1 / 2 ) ( − 1 / 2 ) ( − 3 / 2 ) 3 ! x 3 + ⋯ = 1 + 1 2 x − 1 2 ! 1 2 2 x 2 + 1 3 ! 1 · 3 2 3 x 3 − ⋯ + ( −1 ) n + 1 n ! 1 · 3 · 5 ⋯ ( 2 n − 3 ) 2 n x n + ⋯ = 1 + ∑ n = 1 ∞ ( −1 ) n + 1 n ! 1 · 3 · 5 ⋯ ( 2 n − 3 ) 2 n x n .
  2. From the result in part a. the third-order Maclaurin polynomial is
    p 3 ( x ) = 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 .

    Therefore,
    1.5 = 1 + 0.5 ≈ 1 + 1 2 ( 0.5 ) − 1 8 ( 0.5 ) 2 + 1 16 ( 0.5 ) 3 ≈ 1.2266.

    From Taylor’s theorem, the error satisfies
    R 3 ( 0.5 ) = f ( 4 ) ( c ) 4 ! ( 0.5 ) 4

    for some c between 0 and 0.5 . Since f ( 4 ) ( x ) = − 15 2 4 ( 1 + x ) 7 / 2 , and the maximum value of | f ( 4 ) ( x ) | on the interval ( 0 , 0.5 ) occurs at x = 0 , we have
    | R 3 ( 0.5 ) | ≤ 15 4 ! 2 4 ( 0.5 ) 4 ≈ 0.00244 .

    The function and the Maclaurin polynomial p 3 are graphed in [link] .
    This graph has two curves. The first one is f(x)= the square root of (1+x) and the second is psub3(x). The curves are very close at y = 1.
    The third-order Maclaurin polynomial p 3 ( x ) provides a good approximation for f ( x ) = 1 + x for x near zero.
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Find the binomial series for f ( x ) = 1 ( 1 + x ) 2 .

∑ n = 0 ∞ ( −1 ) n ( n + 1 ) x n

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Common functions expressed as taylor series

At this point, we have derived Maclaurin series for exponential, trigonometric, and logarithmic functions, as well as functions of the form f ( x ) = ( 1 + x ) r . In [link] , we summarize the results of these series. We remark that the convergence of the Maclaurin series for f ( x ) = ln ( 1 + x ) at the endpoint x = 1 and the Maclaurin series for f ( x ) = tan −1 x at the endpoints x = 1 and x = −1 relies on a more advanced theorem than we present here. (Refer to Abel’s theorem for a discussion of this more technical point.)

Maclaurin series for common functions
Function Maclaurin Series Interval of Convergence
f ( x ) = 1 1 − x ∑ n = 0 ∞ x n −1 < x < 1
f ( x ) = e x ∑ n = 0 ∞ x n n ! − ∞ < x < ∞
f ( x ) = sin x ∑ n = 0 ∞ ( −1 ) n x 2 n + 1 ( 2 n + 1 ) ! − ∞ < x < ∞
f ( x ) = cos x ∑ n = 0 ∞ ( −1 ) n x 2 n ( 2 n ) ! − ∞ < x < ∞
f ( x ) = ln ( 1 + x ) ∑ n = 0 ∞ ( −1 ) n + 1 x n n −1 < x ≤ 1
f ( x ) = tan −1 x ∑ n = 0 ∞ ( −1 ) n x 2 n + 1 2 n + 1 −1 < x ≤ 1
f ( x ) = ( 1 + x ) r ∑ n = 0 ∞ ( r n ) x n −1 < x < 1

Earlier in the chapter, we showed how you could combine power series to create new power series. Here we use these properties, combined with the Maclaurin series in [link] , to create Maclaurin series for other functions.
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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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