Math 2260 Exam #3 Practice Problems
1. Does the following series converge or diverge? Explain your answer.
∞
X
2n
.
3n + n3
n=0
2. Does the following series converge or diverge? Explain your answer.
∞
X
n
.
n
3
n=1
3. Does the following series converge or diverge? Explain your answer.
∞
X
1
2n sin
.
n
n=1
4. Does the following series converge or diverge? If it converges, find the sum. If it diverges, explain why.
∞
X
2n + 3 n
.
4n
n=1
5. Find the interval of convergence of the power series
∞
X
(2x − 5)n
.
n2 3 n
n=1
6. Use the first two non-zero terms of an appropriate Taylor series to approximate
Z 1
sin(x2 ) dx.
0
Estimate the error of your approximation (i.e. the difference between your answer and the actual value
of the integral).
7. Find the radius of convergence of the Taylor series
√
∞
X
(−1)n 1 + n
(x − 2)n .
n2
n=2
√
8. Write the second-degree
Taylor polynomial for f (x) = x centered at c = 25. Use this polynomial to
√
approximate 26 and estimate the error of this approximation. (This problem is a bit computationally
intensive for an actual exam problem, but it’s worth working through to see if you understand how
these types of problems work.)
9. Does the series
∞ √
n
X
n
2
n
n=1
converge or diverge. Be sure to give a complete explanation.
1
10. Does the following series converge or diverge? Explain your answer.
∞
X
n!(n + 1)!
.
(3n)!
n=1
11. Does the sequence
∞
n2
arctan
n2 + 1
n=1
converge or diverge? If it converges, find the limit; if it diverges, explain why.
12. Does the series
∞
X
n=2
n2
1
√
− n
converge or diverge? Explain your answer.
13. Does the series
∞
X
n!
n5
n=1
converge or diverge? Explain your answer.
14. Does the series
∞
X
3n
n3
n=1
converge or diverge? Explain your answer.
15. Does the series
∞
X
2n + 3
2 + 3n + 6)2
(n
n=0
converge or diverge? Explain your answer.
16. For which values of x does the series
∞
X
(x − 4)n
5n
n=0
converge? What is the sum of the series when it converges?
17. Does the series
∞
X
(−1)n (n2 + n3 )
n4 + 1
n=1
converge absolutely, converge conditionally, or diverge? Explain your answer.
18. Does the series
∞
X
(2n)!
n (n!)2
2
n=1
converge or diverge? Explain your answer.
19. Does the series
∞
X
(−1)n
n ln(n)
n=2
converge absolutely, converge conditionally, or diverge? Explain your answer.
2
20. For what values of p does the series
∞
X
n=0
(n2
1
+ 1)p
converge? Explain your answer.
21. What is the interval of convergence of the following power series? Explain your answer.
∞
X
3n (x − 2)n
.
n2
n=1
22. What are the first four nonzero terms of the Taylor series centered at x = 0 for the function f (x) = xe3x ?
R 1/2
23. Which of the following gives the value of 0 cos(x2 ) dx correct to within 0.0001 (i.e. within 1/10, 000)?
Explain your answer.
(a)
1
2
1
1
(b) −
2 120
(c)
1
1
−
24 2688
1
1
(d) −
2 320
1
1
(e) −
2 384
24. What is the limit of the following sequence? Explain your answer.
∞
sin(arctan(ln(n)))
n=2
25. Does the series
√
∞
X
n=2
n2
n
−n+1
converge or diverge? Explain your answer.
26. Does the series
∞
X
(−1)n (n2 + n3 )
n4 + ln(n)
n=1
converge absolutely, converge conditionally, or diverge? Explain your answer.
27. If it converges, find the sum of the series
∞
X
(−1)n
.
n!2n
n=0
If the series diverges, explain why.
28. For which values of p does the following series converge? Explain your answer.
∞
X
√
1 + np .
n=1
29. What is the interval of convergence of the following series? Explain your answer.
∞
X
(x − 1)n
.
n2
n=1
3
(f)
1
1
−
24 1024