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8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
Activity 8.23.
In Activity 8.22 we determined small order Taylor polynomials for a few familiar
functions, and also found general patterns in the derivatives evaluated at 0. Use that
information to write the Taylor series centered at 0 for the following functions.
(a) f (x) =
1
1−x
(b) f (x) = cos(x) (You will need to carefully consider how to indicate that many of
the coefficients are 0. Think about a general way to represent an even integer.)
(c) f (x) = sin(x) (You will need to carefully consider how to indicate that many of
the coefficients are 0. Think about a general way to represent an odd integer.)
⊳
The next activity further considers the important issue of the x-values for which the
Taylor series of a function converges to the function itself.
Activity 8.24.
(a) Plot the graphs of several of the Taylor polynomials centered at 0 (of order at
least 5) for e x and convince yourself that these Taylor polynomials converge to
e x for every value of x.
(b) Draw the graphs of several of the Taylor polynomials centered at 0 (of order at
least 6) for cos(x) and convince yourself that these Taylor polynomials converge
to cos(x) for every value of x. Write the Taylor series centered at 0 for cos(x).
(c) Draw the graphs of several of the Taylor polynomials centered at 0 for
1
1−x .
Based on your graphs, for what values of x do these Taylor polynomials appear
to converge to
1
1−x ? How is this situation different from what we observe with
e x and cos(x)? In addition, write the Taylor series centered at 0 for
1
1−x .
⊳
The Maclaurin series for e x , sin(x), cos(x), and
1
1−x will be used frequently, so we
should be certain to know and recognize them well.
The Interval of Convergence of a Taylor Series
In the previous section (in Figure 8.6 and Activity 8.24) we observed that the Taylor
polynomials centered at 0 for e x , cos(x), and sin(x) converged to these functions for
all values of x in their domain, but that the Taylor polynomials centered at 0 for
1
1−x
converged to
1
1−x for only some values of x. In fact, the Taylor polynomials centered at 0
for
1
1−x converge to
1
1−x on the interval (−1, 1) and diverge for all other values of x. So
the Taylor series for a function f (x) does not need to converge for all values of x in the
domain of f .
Our observations to date suggest two natural questions: can we determine the values
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
Activity 8.23.
In Activity 8.22 we determined small order Taylor polynomials for a few familiar
functions, and also found general patterns in the derivatives evaluated at 0. Use that
information to write the Taylor series centered at 0 for the following functions.
(a) f (x) =
1
1−x
(b) f (x) = cos(x) (You will need to carefully consider how to indicate that many of
the coefficients are 0. Think about a general way to represent an even integer.)
(c) f (x) = sin(x) (You will need to carefully consider how to indicate that many of
the coefficients are 0. Think about a general way to represent an odd integer.)
⊳
The next activity further considers the important issue of the x-values for which the
Taylor series of a function converges to the function itself.
Activity 8.24.
(a) Plot the graphs of several of the Taylor polynomials centered at 0 (of order at
least 5) for e x and convince yourself that these Taylor polynomials converge to
e x for every value of x.
(b) Draw the graphs of several of the Taylor polynomials centered at 0 (of order at
least 6) for cos(x) and convince yourself that these Taylor polynomials converge
to cos(x) for every value of x. Write the Taylor series centered at 0 for cos(x).
(c) Draw the graphs of several of the Taylor polynomials centered at 0 for
1
1−x .
Based on your graphs, for what values of x do these Taylor polynomials appear
to converge to
1
1−x ? How is this situation different from what we observe with
e x and cos(x)? In addition, write the Taylor series centered at 0 for
1
1−x .
⊳
The Maclaurin series for e x , sin(x), cos(x), and
1
1−x will be used frequently, so we
should be certain to know and recognize them well.
The Interval of Convergence of a Taylor Series
In the previous section (in Figure 8.6 and Activity 8.24) we observed that the Taylor
polynomials centered at 0 for e x , cos(x), and sin(x) converged to these functions for
all values of x in their domain, but that the Taylor polynomials centered at 0 for
1
1−x
converged to
1
1−x for only some values of x. In fact, the Taylor polynomials centered at 0
for
1
1−x converge to
1
1−x on the interval (−1, 1) and diverge for all other values of x. So
the Taylor series for a function f (x) does not need to converge for all values of x in the
domain of f .
Our observations to date suggest two natural questions: can we determine the values