Practice Problems (Taylor and Maclaurin Series)
1. By definition, the Maclaurin series for a function f (x) is given by
f (x) =
∞
X
f 00 (0) 2 f 000 (0) 3
f (n) (0) n
x = f (0) + f 0 (0)x +
x +
x + ···
n!
2!
3!
n=0
Use this definition directly to determine the Maclaurin series for each function and find its
interval of convergence.
1
1−x
(b) ex
(a)
(c) sin x
(d) cos x
(e) ln(1 + x)
(f) tan−1 x
(g) (1 + x)k
2. Now that you’ve derived the Maclaurin series for the important functions listed above, you
should memorize them.
3. Determine whether the following series converge or diverge. Find the exact sum of those series
which converge. To find the exact sum, it may help to think about some of the important
Maclaurin series.
23 25 27 29 211
+ 2 + 3 + 4 + 5 + ···
(a)
3
3
3
3
3
22 23 24
+
+
+ ···
2!
3!
4!
1 1 1 1
1
(c) 1 − + − + −
+ ···
3 5 7 9 11
1
1
1
1
1
(d) 1 − + − + −
+ ···
3! 5! 7! 9! 11!
(b) 1 + 2 +
(e) 1 + 2(0.5) + 3(0.5)2 + 4(0.5)3 + 5(0.5)4 + 6(0.5)5 + · · ·
1
1
1
1
+ + + + ···
2! 3! 4! 5!
1 1 1 1 1
(g) 1 − + − + − + · · ·
2 3 4 5 6
1
2
3
4
5
(h) + 2 + 3 + 4 + 5 + · · ·
3 3
3
3
3
(f) 1 + 1 +
4. Determine the Maclaurin series and interval of convergence for each function. You should
try to use substitution, addition, subtraction, multiplication, division, differentiation, and
integration of known Maclaurin series instead of determining the series from the definition.
(a)
1
1 + 2x
(b)
3x
1 − x2
(c) x sin x2
(d) e−2x
(e) e2+x
√
(f) ex
(g)
6
2+x
(h) ln (2 + 2x)5
(i)
1+x
1−x
1−x
(j) ln
1+x
(k)
5x − 1
x2 − 1
(l)
1
(1 − x)2
(m) ex sin x
2
(n) ex cos x
(o) 6 sin x cos x
(p)
tan−1 x
1+x
(q)
ln (1 + x)
1−x
(r) x2 e−x
(s)
x
(1 − x)(1 − 2x)
5. Find the exact sum of the series
∞
X
k(k − 1)xk−2 by recognizing how it is related to something
k=2
familiar. For which values of x is this sum valid?
6. Use Maclaurin series to approximate the following definite integrals. Give an upper bound on
the error in your approximation.
(a)
Z 1
2
e−x dx
0
(b)
Z 1
sin x2 dx
0
(c)
Z 1
0
8
dx
8 + x3
7. Use Maclaurin series to evaluate the following limits.
sin x
x→0 x
(a) lim
sin x − x + x3 /6
x→0
x5
(b) lim
tan−1 x − x
x→0
x3
(c) lim
8. Approximate ln (1.1) with an error of less than 0.001 by looking at Taylor polynomials based
at 1 for ln x. How do you know that the error is less than 0.001?
1
9. Approximate √ with an error of less than 0.1 by using an appropriate Maclaurin polynomial.
e
How do you know that the error is less than 0.1?
10. Use an appropriate Taylor polynomial of degree 2 to obtain an estimate for
11. Use an appropriate Taylor polynomial of degree 2 to obtain an estimate for
√
3
√
1.3.
101.
12. Use the Maclaurin polynomial of degree 4 for cos x to approximate cos (1). Without using
your calculator, give a bound for the error in your approximation.
13. Use the Maclaurin polynomial of degree 4 for cos x to approximate cos (100). Without using
your calculator, give a bound for the error in your approximation. Can you think of any way
to improve on the error bound?
14. Find the Maclaurin series for f (x) = 4 arctan x. What interesting fact do you learn when
using the series to approximate f (1) ?