AP Calculus Practice Problems:
Series, Taylor/Maclaurin, Error Bounds, and
Convergence
Practice Questions
1. Determine whether →
! 1
n=1
2n
converges or diverges.
2. Find the sum of → " #n
! 1
.
n=0
3
3. Does →
! 1
n=1
n
converge or diverge?
4. Test →
! 1
n=1
n2
for convergence.
5. Determine whether →
! (→1)n
n=1
n
converges absolutely, conditionally, or diverges.
6. Approximate
→
! (→1)n
n=1
n
using the first 5 terms. Bound the error.
2
, 7. Apply the Nth Term Test to
→
! n
.
n=1
n+1
8. Use the Ratio Test on →
! n!
n
.
n=1
3
9. Apply the Limit Comparison Test to
→
! 1
n=1
n2 +1
using
!→
1
2
.
n=1
n
10. Determine whether →
! (→1)n
↑
n=1
n
converges absolutely, conditionally, or diverges.
11. Find the Maclaurin polynomial of degree 3 for
f (x) = ex .
12. Approximate e0.1 using the degree 3 Maclaurin polynomial. Estimate the error.
13. Construct the Taylor polynomial of degree 2 for
f (x) = ln(x)
centered at c = 1.
14. Approximate ln(1.1) using your polynomial.
15. Find the Maclaurin polynomial of degree 4 for sin x.
16. Approximate sin(0.2) using the degree 4 polynomial. Bound the error.
17. Find the Maclaurin polynomial of degree 4 for cos x.
18. Approximate cos(0.2) using the degree 4 polynomial. Bound the error.
19. Construct the degree 3 Taylor polynomial for
↑
f (x) = x
centered at c = 4.
3
, ↑
20. Approximate 4.1 using your polynomial.
21. Use the Lagrange Error Bound to estimate the error in approximating ex at x = 0.5
with the degree 2 Maclaurin polynomial.
22. Use the Alternating Series Error Bound for
→
! (→1)n
n=1
n2
truncated after 5 terms.
23. Estimate the error in approximating sin(1) with the degree 3 Maclaurin polynomial.
24. Estimate the error in approximating cos(1) with the degree 2 Maclaurin polynomial.
25. Bound the error in approximating ln(1.2) with the degree 2 Taylor polynomial centered
at c = 1.
26. Expand
1
1→x
as a Maclaurin series up to x4 .
27. Expand ex up to x4 .
28. Expand sin x up to x5 .
29. Expand cos x up to x4 .
30. Use a series expansion to approximate e0.2 .
31. Find the interval of convergence for
→
! xn
.
n=1
n
32. Find the interval of convergence for
→
! xn
.
n=0
2n
33. Find the interval of convergence for
→
! (→1)n xn
.
n=1
n
4
Series, Taylor/Maclaurin, Error Bounds, and
Convergence
Practice Questions
1. Determine whether →
! 1
n=1
2n
converges or diverges.
2. Find the sum of → " #n
! 1
.
n=0
3
3. Does →
! 1
n=1
n
converge or diverge?
4. Test →
! 1
n=1
n2
for convergence.
5. Determine whether →
! (→1)n
n=1
n
converges absolutely, conditionally, or diverges.
6. Approximate
→
! (→1)n
n=1
n
using the first 5 terms. Bound the error.
2
, 7. Apply the Nth Term Test to
→
! n
.
n=1
n+1
8. Use the Ratio Test on →
! n!
n
.
n=1
3
9. Apply the Limit Comparison Test to
→
! 1
n=1
n2 +1
using
!→
1
2
.
n=1
n
10. Determine whether →
! (→1)n
↑
n=1
n
converges absolutely, conditionally, or diverges.
11. Find the Maclaurin polynomial of degree 3 for
f (x) = ex .
12. Approximate e0.1 using the degree 3 Maclaurin polynomial. Estimate the error.
13. Construct the Taylor polynomial of degree 2 for
f (x) = ln(x)
centered at c = 1.
14. Approximate ln(1.1) using your polynomial.
15. Find the Maclaurin polynomial of degree 4 for sin x.
16. Approximate sin(0.2) using the degree 4 polynomial. Bound the error.
17. Find the Maclaurin polynomial of degree 4 for cos x.
18. Approximate cos(0.2) using the degree 4 polynomial. Bound the error.
19. Construct the degree 3 Taylor polynomial for
↑
f (x) = x
centered at c = 4.
3
, ↑
20. Approximate 4.1 using your polynomial.
21. Use the Lagrange Error Bound to estimate the error in approximating ex at x = 0.5
with the degree 2 Maclaurin polynomial.
22. Use the Alternating Series Error Bound for
→
! (→1)n
n=1
n2
truncated after 5 terms.
23. Estimate the error in approximating sin(1) with the degree 3 Maclaurin polynomial.
24. Estimate the error in approximating cos(1) with the degree 2 Maclaurin polynomial.
25. Bound the error in approximating ln(1.2) with the degree 2 Taylor polynomial centered
at c = 1.
26. Expand
1
1→x
as a Maclaurin series up to x4 .
27. Expand ex up to x4 .
28. Expand sin x up to x5 .
29. Expand cos x up to x4 .
30. Use a series expansion to approximate e0.2 .
31. Find the interval of convergence for
→
! xn
.
n=1
n
32. Find the interval of convergence for
→
! xn
.
n=0
2n
33. Find the interval of convergence for
→
! (→1)n xn
.
n=1
n
4
