4.4. TAYLOR SERIES 165
Problem 4.90 Find the Taylor expansion of f .x/ D cos.x/ using a center of c D =4. Use the
formula for the Taylor expansion. Warning: this is a little messy.
Problem 4.91 Using the Taylor series for sin.x/, cos.x/ and e
x
, prove Euler’s identity:
e
i
D i sin.x/ C cos.x/
Problem 4.92
For each of the series you found in Problem
4.89, find the radius and interval
of convergence.
Problem4.93 Prove that the Taylor series for a polynomial function p.x/ is just the polynomial
itself.
Problem 4.94 Find a power series expansion for
f .x/ D
1
x
2
3x C 2
Problem 4.95 Find a power series expansion for
f .x/ D
1
4 4x C x
2
Problem 4.96 Find a power series expansion for
f .x/ D
1
x
3
6x
2
C 11x C 6
Problem 4.97 Find a power series expansion for
f .x/ D
1
x
3
C x
166 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
Problem 4.98 If
f .x/ D p.x/e
x
where p.x/ is a polynomial, demonstrate that f .x/ has a power series expansion with radius of
convergence r D 1.
Problem 4.99 Find the Taylor polynomial of degree n for the given function with the given
center c.
1. f .x/ D cos.x/ for n D 6 at c D 0,
2. g.x/ D sin.2x/ for n D 7 at c D 0,
3. h.x/ D e
x
for n D 5 at c D 0,
4. r.x/ D log.x/ for n D 3 at c D 1,
5. s.x/ D tan
1
.x/ for n D 8 at c D 0,
6. q.x/ D x
2
C 3x C 5 for n D 2 at c D 1.
Problem 4.100 Suppose we have T
5
.x/ for f .x/ D e
x
at c D 0. Compute a bound on the size
of R
5
.x/ with Taylor’s inequality.
Problem 4.101 Find the smallest n for which T
n
.x/ on f .x/ D cos.x/ has jR
n
.x/j < 0:01 on
3 x 3 with c D 0.
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