Maclaurin Series
Practice Problems and Solutions
Practice Problems
1. Find the Maclaurin series for f (x) = e2x up to the x4 term.
2. Approximate sin(0.1) using the first three nonzero terms of its Maclaurin series.
3. Derive the Maclaurin series for cos(x2 ) up to x6 .
4. Use the Maclaurin series of ex to approximate e0.5 with error less than 0.001.
5. Show that differentiating ∞ xn
P
n=0 n! term by term reproduces the same series.
6. Find the Maclaurin series for ln(1 + x) up to x4 .
7. Approximate cos(0.2) using four terms of its Maclaurin series.
sin x
8. Derive the Maclaurin series for x
.
9. Use the Maclaurin series of e−x to approximate e−1 .
10. Show that sin(x2 ) has only odd powers of x in its expansion.
1
11. Find the Maclaurin series for 1−x
and state its radius of convergence.
12. Approximate tan−1 (0.5) using three terms of its Maclaurin series.
13. Derive the Maclaurin series for cosh(x).
14. Show that sinh(x) and cosh(x) differ only in the signs of odd terms.
15. Approximate ln(1.1) using the Maclaurin expansion of ln(1 + x).
16. Find the Maclaurin series for sinx x up to x6 .
√
17. Approximate 1 + x using its binomial series expansion up to x3 .
18. Show that the Maclaurin series for cos(x) converges for all real x.
1
19. Derive the Maclaurin series for 1+x2
.
20. Approximate sin(0.5) using four terms of its Maclaurin series.
1
Practice Problems and Solutions
Practice Problems
1. Find the Maclaurin series for f (x) = e2x up to the x4 term.
2. Approximate sin(0.1) using the first three nonzero terms of its Maclaurin series.
3. Derive the Maclaurin series for cos(x2 ) up to x6 .
4. Use the Maclaurin series of ex to approximate e0.5 with error less than 0.001.
5. Show that differentiating ∞ xn
P
n=0 n! term by term reproduces the same series.
6. Find the Maclaurin series for ln(1 + x) up to x4 .
7. Approximate cos(0.2) using four terms of its Maclaurin series.
sin x
8. Derive the Maclaurin series for x
.
9. Use the Maclaurin series of e−x to approximate e−1 .
10. Show that sin(x2 ) has only odd powers of x in its expansion.
1
11. Find the Maclaurin series for 1−x
and state its radius of convergence.
12. Approximate tan−1 (0.5) using three terms of its Maclaurin series.
13. Derive the Maclaurin series for cosh(x).
14. Show that sinh(x) and cosh(x) differ only in the signs of odd terms.
15. Approximate ln(1.1) using the Maclaurin expansion of ln(1 + x).
16. Find the Maclaurin series for sinx x up to x6 .
√
17. Approximate 1 + x using its binomial series expansion up to x3 .
18. Show that the Maclaurin series for cos(x) converges for all real x.
1
19. Derive the Maclaurin series for 1+x2
.
20. Approximate sin(0.5) using four terms of its Maclaurin series.
1
