Some Practice Problems

More problems can be found in section 5.2 of Strang's book.

Compute each of the indicated indefinite integrals.

1. \( \indef{\sin(12x)} \)
2. \( \indef{x \tan(x^2)} \)
3. \( \indef{\sin^2(x)\cos(x)} \)
4. \( \indef{\exp(5x)} \)
5. \( \indef{\frac{\ln(x)}{x}} \)
6. \( a^x \) where \( 0 < a \neq 1 \) is an unknown constant.

Compute each of the indicated definite integrals.

1. \( \defint{-50}{75}{\cos(\pi x)} \)
2. \( \defint{0}{\tfrac{\pi}{6}}{\tan(x)} \)
3. \( \defint{\tfrac{\pi}{2}}{\pi}{\sin(x)\cos(x)} \)
4. \( \defint{-1}{1}{5x^2\exp(2x^3)} \)
5. \( \defint{0}{2}{\frac{x^2\ln(x^3 + 1)}{x^3 + 1}} \)

More problems can be found in section 7.1 of Strang's book.

Compute each of the following indefinite integrals.

1. \( \indef{x\sin(x)} \)
2. \( \indef{x\exp(4x)} \)
3. \( \indef{x\exp(-x)} \)
4. \( \indef{x\cos(5x)} \)
5. \( \indef{x^2\cos(3x)} \)
6. \( \indef{x^2\exp(7x)} \)
7. \( \indef{\ln(x)} \)
8. \( \indef{x\ln(x)} \)
9. \( \indef{x^{-1}\ln(x)} \)
10. \( \indef{\ln(2x+1)} \)
11. \( \indef{x^3 e^x} \)
12. \( \indef{e^x \sin(x)} \)
13. \( \indef{e^x \cos(x)} \)
14. \( \indef{\sin(\ln(x))} \)
15. \( \indef{\cos(\ln(x))} \)
16. \( \indef{x \sec^2(x)} \)
17. \( \indef{x\operatorname{arcsec}{x}} \)
18. \( \indef{x^4\arctan{x}} \)
19. \( \indef{\arcsin(x)} \)

Compute each of the following indefinite integrals. These ones can be a bit more involved or tricky…

1. \( \indef{x \sin(x) e^x} \)
2. \( \indef{x \cos(x) e^x} \)
3. \( \indef{x^n \ln(x)} \) where \( n \neq -1 \) is an unknown real number.
4. \( \indef{\exp(\alpha x) \cos(\beta x)} \) where \( \alpha, \beta \) are unknown constants with \( \alpha \neq 0 \neq \beta \).
5. \( \indef{\exp(\alpha x) \sin(\beta x)} \) where \( \alpha, \beta \) are unknown constants with \( \alpha \neq 0 \neq \beta \).
6. \( \log_a(x) \) where \( 0 < a \neq 1 \) is an unknown constant.

Compute each of the following definite integrals.

1. \( \defint{1}{2}{\ln(x)} \)
2. \( \defint{1}{e}{\ln(x^3)} \)
3. \( \defint{-\pi}{\pi}{x^2\sin(x)} \)
4. \( \defint{0}{1}{x\exp(-x^2)} \)
5. \( \defint{0}{3}{\ln(x^2 + 1)} \)
6. \( \defint{-\pi}{\pi}{x\sin(x)} \)
7. \( \defint{1}{e}{x\ln(x)} \)
8. \( \defint{3}{5}{\ln(2x+1)} \)

Find a "power reduction formula" for each of the following integrals using Integration by parts. Your final answer should be a formula with smaller powers on the right hand side. (Each of these can be used to obtain sum-type formulas for the indefinite integrals in question. As an extra "challenge problem", find that sum-type formula; they can be quite tricky…)

1. \( \indef{x^n e^x} \)
2. \( \indef{x^n \cos(x)} \)
3. \( \indef{x^n \sin(x)} \)
4. \( \indef{(\ln(x))^n} \)
5. \( \indef{x (\ln(x))^n} \)
6. \( \indef{x^k (\ln(x))^n} \)

More problems can be found in section 7.2 of Strang's book.

Compute the following integrals (for practice with definite integrals, use bounds \( a = \tfrac{\pi}{6} \) and \( b = \tfrac{5\pi}{3} \)).

1. \( \indef{\cos^2(x)} \)
2. \( \indef{\sin^2(x)} \)
3. \( \indef{\sin^2(x)\cos(x)} \)
4. \( \indef{\cos^{101}(x)} \)
5. \( \indef{\sin^8(x)\cos^4(x)} \)
6. \( \indef{\cos(x)\sin^11(x)} \) (do this in two different ways).
7. \( \indef{\cos^4(x)\sin^2(x)} \)
8. \( \indef{\cos^2(x)\sin^4(x)} \)
9. \( \indef{\cos(2x)\sin(x)} \) (hint: what can you do to make that better?).

Compute the following integrals (for practice with definite integrals, use bounds \( a = -\tfrac{\pi}{6} \) and \( b = \tfrac{\pi}{4} \)).

1. \( \indef{\sec^2(x)\tan^5(x)} \) (do this in two different ways)
2. \( \indef{\sec^4(x)\tan^{10}(x)} \)
3. \( \indef{\sec^3(x)\tan(x)} \)
4. \( \indef{\sec^2(x)\tan^4(x)} \)
5. \( \indef{\sec(x)\tan^2(x)} \)
6. \( \indef{\tan(x)} \)
7. \( \indef{\sec(x)} \)
8. \( \indef{\tan^2(x)} \) (hint: you need partial fractions)

Compute the following indefinite integrals (for practice with definite integrals, use bounds \( a = \tfrac{\pi}{6} \) and \( b = \tfrac{3\pi}{4} \)).

1. \( \indef{\csc^2(x)\cot^5(x)} \) (do this in two different ways)
2. \( \indef{\csc^4(x)\cot^{10}(x)} \)
3. \( \indef{\csc^3(x)\cot(x)} \)
4. \( \indef{\csc^2(x)\cot^4(x)} \)
5. \( \indef{\csc(x)\cot^2(x)} \)
6. \( \indef{\cot(x)} \)
7. \( \indef{\csc(x)} \)
8. \( \indef{\cot^2(x)} \) (hint: you need partial fractions)

More problems can be found in section 7.3 of Strang's book.

Compute each of the following indefinite integrals.

1. \( \indef{\frac{1}{\sqrt{9 - x^2}}} \)
2. \( \indef{\frac{1}{\sqrt{x^2 - 49}}} \)
3. \( \indef{\sqrt{4 - x^2}} \)
4. \( \indef{\frac{1}{1 + 9x^2}} \)
5. \( \indef{\frac{x^2}{\sqrt{1 - x^2}}} \)
6. \( \indef{\frac{1}{(1 + x^2)^3}} \)
7. \( \indef{\frac{\sqrt{x^2 - 16}}{x}} \)
8. \( \indef{\frac{x^3}{\sqrt{9 - x^2}}} \)
9. \( \indef{\frac{1}{\sqrt{x^6 - x^4}}} \)
10. \( \indef{\sqrt{x^6 - x^8}} \)
11. \( \indef{\frac{1}{1 + x^2}^{3/2}} \)
12. \( \indef{\frac{1}{(1 - 7x + x^2)}} \)
13. \( \indef{\frac{1}{x^2 + 4x - 5}^{3/2}} \)
14. \( \indef{\frac{x^2}{\sqrt{x^2 - 1}}} \)
15. \( \indef{\frac{x^2}{x^2 + 6x + 25}} \)
16. \( \indef{\frac{1}{x^2\sqrt{x^2 + 1}}} \)
17. \( \indef{\frac{x^2}{\sqrt{1 + x^2}}} \)

Compute each of the following definite integrals.

1. \( \defint{-1}{1}{(1 - x^2)^{3/2}} \)
2. \( \defint{0}{1/2}{\frac{1}{\sqrt{1 - x^2}}} \)
3. \( \defint{1}{4}{\frac{1}{\sqrt{x^2 - 1}}} \)
4. \( \defint{-1}{1}{\frac{x}{x^2 + 1}} \)
5. \( \defint{-3}{3}{\frac{1}{x^2 + 8}} \)
6. \( \defint{-a}{a}{\sqrt{a^2 - x^2}} \)

Compute each of the following indefinite integrals, where \( a, b > 0 \) are unknown constants.

1. \( \indef{\sqrt{a^2 + (bx)^2}} \)
2. \( \indef{\sqrt{a^2 - (bx)^2}} \)
3. \( \indef{\sqrt{(bx)^2 - a^2}} \)
4. \( \indef{\frac{1}{\sqrt{a^2 + (bx)^2}}} \)
5. \( \indef{\frac{1}{\sqrt{a^2 - (bx)^2}}} \)
6. \( \indef{\frac{1}{\sqrt{(bx)^2 - a^2}}} \)

Compute each of the following indefinite integrals using an inverse trigonometric substitution. Then compute the same integral "the normal way". Which do you prefer, and why?

1. \( \indef{1 + x^2} \)
2. \( \indef{1 - x^2} \)
3. \( \indef{x^2 - 1} \)

More problems can be found in section 7.4 of Strang's book.

Compute each of the following indefinite integrals.

1. \( \indef{\frac{1}{x^2 - 5x + 6}} \)
2. \( \indef{\frac{x}{x^2 - 5x + 6}} \)
3. \( \indef{\frac{1}{x(x - 1)}} \)
4. \( \indef{\frac{1}{x(x - 1)(x - 2)}} \)
5. \( \indef{\frac{1}{x^2(x - 1)(x - 2)}} \)
6. \( \indef{\frac{3x + 1}{x^2}} \)
7. \( \indef{\frac{3x^2}{x^2 + 1}} \)
8. \( \indef{\frac{x}{(x - 1)(x^2 + 1)^2}} \)
9. \( \indef{\frac{x}{x^2 - 4}} \)
10. \( \indef{\frac{1}{x^2(x - 1)}} \)
11. \( \indef{\frac{x^2 + 1}{x + 1}} \)
12. \( \indef{\frac{1}{x^4 - 1}} \)
13. \( \indef{\frac{1}{\sqrt[3]{x} + \sqrt{x}}} \)
14. \( \indef{\frac{1}{1 + \sqrt{x + 1}}} \)
15. \( \indef{\frac{1 + e^x}{1 - e^x}} \)

For this section, the following is referred to as the "sequence table".

1. \( a_n = 3^n7^{-n} \)
2. \( a_n = e^{-\frac{1}{\sqrt{n}}} \)
3. \( a_n = \sqrt{\frac{1+4n^2}{1+n^2}} \)
4. \( a_n = \frac{n^2}{\sqrt{n^3+4n}} \)
5. \( a_n = \sin(n) \)
6. \( a_n = \sqrt[n]{n} \)
7. \( a_n = n-\sqrt{n+1}\sqrt{n+3} \)
8. \( a_n = \frac{n!}{2^n} \)
9. \( a_n = \frac{(-3)^n}{n!} \)
10. \( a_n = \cos(n) \)
11. \( a_n = \frac{1}{2n+3} \)
12. \( a_n = \frac{1-n}{2+n} \)
13. \( a_n = n(-1)^n \)
14. \( a_n = 2+\frac{(-1)^n}{n} \)
15. \( a_n = 3-2ne^{-n} \)
16. \( a_n = n^3-3n+3 \)
17. \( a_n = \frac{4^n}{(n+1)!} \)
18. \( a_n = \left(-\frac{3}{2}\right)^n \)
19. \( a_n = -\frac{n^{n+1}}{n+2} \)
20. \( a_n = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right) \)
21. \( a_n = 3n+1 \)
22. \( a_n = (-1)^{n+1}\frac{3}{2^{n-1}} \)
23. \( a_n = 10\cdot2^{n-1} \)
24. \( a_n = 1/(n-1)! \)
25. \( a_n = (-1)^n\frac{n}{n+1} \)
26. \( a_n = \frac{4n^2-n+5}{3n^2+1} \)
27. \( a_n = \frac{4^n}{5^n} \)
28. \( a_n = \frac{n-1}{n}-\frac{n}{n-1},n\geq2 \)
29. \( a_n = \ln(n) \)
30. \( a_n = \frac{3n}{\sqrt{n^2+1}} \)
31. \( a_n = \left(1+\frac{1}{n}\right)^n \)
32. \( a_n = \sin(n) \)
33. \( a_n = \tan(n) \)
34. \( a_n = (-1)^n\frac{3n-1}{n} \)
35. \( a_n = \frac{3n^2-1}{n} \)
36. \( a_n = n\cos{n} \)
37. \( a_n = 2^n-n! \)
38. \( a_n = 5-\frac{1}{n} \)
39. \( a_n = \frac{(-1)^{n+1}}{n} \)
40. \( a_n = \frac{1.1^n}{n} \)
41. \( a_n = \frac{2n}{n+1} \)
42. \( a_n = (-1)^n\frac{n^2}{2^n-1} \)
43. \( a_n = \frac{n}{n+2} \)
44. \( a_n = \frac{n^2-6n+9}{n} \)
45. \( a_n = (-1)^n\frac{1}{n^3} \)
46. \( a_n = \frac{n^2}{2^n} \)
47. \( a_n = \frac{2^n-20}{7\cdot2^n} \)
48. \( a_n = \sin(3/n)\left(1+\frac{2}{n}\right)^n \)
49. \( a_n = \left(1+\frac{2}{n}\right)^{2n} \)

For each sequence in the sequence table, complete the following exercises. Note that a variety of techniques may be required to do so.

1. Compute the first five terms of the sequence.
2. Decide if the sequence is monotone increasing, monotone decreasing, or neither.
3. Decide if the sequence converges or diverges. If it converges, compute the limit.

4. Decide if the sequence is bounded above, bounded below, both, or neither. If the it is bounded above or below, give bounding values.

   Challenge: Make the bounds as tight as possible.

Complete each of the following.

1. Compute the limit of the sequence defined by \( a_{n+1} = \begin{cases} 1 & \text{if } n = 0 \\ \frac{1}{1+a_n} & \text{otherwise} \end{cases} \).

   Note: You do not need to show that the limit exists as part of your computation.

2. Compute the limit of the sequence defined by \( a_n = \frac{F_{n+1}}{F_n} \) for \( n \geq 1 \) where \(F_n\) is the \(n^{th}\) Fibonacci number.

   Note: You do not need to show that the limit exists as part of your computation.

3. Give an example of two sequences \( (a_n) \) and \( (b_n) \) such that \( a_n < b_n \) for all \( n \), but \( a_n \to L \) and \( b_n \to L \) as \( L \to \infty \).

The following exercises vary in difficulty, and test your understanding of how sequences work at a fundamental level.

Complete each of the following problems using basic concepts from the course (i.e. definitions and theorems).

1. Prove the basic properties of sequential limits using the definition of the limit of a sequence. I.e., if \( (a_n), (b_n) \) are sequences with \( a_n \to A \) and \( b_n \to B \) as \( n \to \infty \), and \( c \) is a constant, prove the following with the definition of the limit.

   Constant Law

   \( c \to c \) as \( n \to \infty \).

   Sum Law

   \( (a_n + b_n) \to A + B \) as \( n \to \infty \).

   Product Law

   \( (a_n \cdot b_n) \to A \cdot B \) as \( n \to \infty \).

   Quotient Law

   \( \left(\frac{a_n}{b_n}\right) \to \frac{A}{B} \) as \( n \to \infty \), provided \( B \neq 0 \).

   Tail Behaviour

   For every \( k \) we have \( a_{n + k} \to A \) as \( n \to \infty \).

   Comparison Law

   If there are \( k \) and \( N \) such that \( a _n \leq b_{n+k} \) for all \( n \geq N \), then \( A \leq B \).

   Squeeze Theorem

   Let \( (a_n) \), \( (b_n) \), \( (c_n) \) be sequences with \( a_n \to A \), \( b_n \to B \), and \( c_n \to C \) as \( n \to \infty \). If \( a_n \leq b_n \leq c_n \) for all \( n \), then \( A \leq B \leq C \).

2. Use the definition of the limit of a sequence to show that if \( \lim_{n\to\infty} |a_n| = 0 \), then \( \lim_{n\to\infty} a_n = 0 \).

For each of the series below, complete each of the following.

• Classify the series using the special types of series we have.
• Determine if the series converges or diverges using convergence tests.
• Compute the sum of the series using known methods (if possible).
• If the series converges, determine if it converge absolutely or conditionally.

1. \( \sum_{n = 1}^\infty \frac{(-1)^n}{n} \)
2. \( \sum_{n = 1}^\infty \frac{1}{n^2} \)
3. \( \sum_{n = 1}^\infty \cos(\pi n) \)
4. \( \sum_{n = 1}^\infty n \)
5. \( \sum_{n = 1}^\infty \frac{1}{n!} \)
6. \( \sum_{n = 1}^\infty \frac{1}{3^n} \)
7. \( \sum_{n = 1}^\infty \left(-\frac{9}{10}\right)^n \)
8. \( \sum_{n = 1}^\infty \left(\frac{1}{10}\right)^n \)
9. \( \sum_{n = 0}^\infty \frac{1}{4^n} \)
10. \( \sum_{n = 0}^\infty n^3 \)
11. \( \sum_{n = 1}^\infty (-1)^n n \)
12. \( \sum_{n = 0}^\infty \frac{5}{2^n} \)
13. \( \sum_{n = 1}^\infty \frac{1}{(2n-1)(2n+1)} \)
14. \( \sum_{n = 1}^\infty \ln \left(\frac{n}{n+1}\right) \)
15. \( \sum_{n = 1}^\infty \frac{n}{(2n-1)(2n+1)} \)
16. \( \sum_{n = 1}^\infty \frac{2n+1}{n^2(n+1)^2} \)
17. \( \sum_{n = 1}^\infty \frac{1}{n(n+1)} \)
18. \( \sum_{n = 1}^\infty \frac{3}{n(n+2)} \)
19. \( \sum_{n = 1}^\infty \frac{3n^2}{n(n+2)} \)
20. \( \sum_{n = 1}^\infty \frac{2^n}{n^2} \)
21. \( \sum_{n = 1}^\infty \frac{n!}{10^n} \)
22. \( \sum_{n = 1}^\infty \frac{5^n-n^5}{5^n+n^5} \)
23. \( \sum_{n = 0}^\infty e^{-n} \)
24. \( \sum_{n = 1}^\infty \frac{2^n+1}{2^{n+1}} \)
25. \( \sum_{n = 1}^\infty \left(1+\frac1n\right)^n \)
26. \( \sum_{n = 2}^\infty \frac{1}{n^2-1} \)
27. \( \sum_{n = 0}^\infty \big(\sin 1\big)^n \)
28. \( \sum_{n = 1}^\infty \frac{1}{n^5} \)
29. \( \sum_{n = 0}^\infty \frac{1}{5^n} \)
30. \( \sum_{n = 0}^\infty \frac{6^n}{5^n} \)
31. \( \sum_{n = 1}^\infty n^{-4} \)
32. \( \sum_{n = 1}^\infty \sqrt{n} \)
33. \( \sum_{n = 0}^\infty \frac{10}{n!} \)
34. \( \sum_{n = 1}^\infty \frac{2}{(2n+8)^2} \)
35. \( \sum_{n = 1}^\infty \frac{1}{2n} \)
36. \( \sum_{n = 1}^\infty \frac{1}{2n-1} \)
37. \( \sum_{n = 1}^\infty 2 \)
38. \( \sum_{n = 1}^\infty n^{-n} \)
39. \( \sum_{n = 1}^\infty n^{\frac{1}{n}} \)
40. \( \sum_{n = 1}^\infty \frac{\ln n}{n^3} \)
41. \( \sum_{n = 1}^\infty \frac{1}{2^n} \)
42. \( \sum_{n = 1}^\infty \frac{1}{n^4} \)
43. \( \sum_{n = 1}^\infty \frac{n}{n^2+1} \)
44. \( \sum_{n = 2}^\infty \frac{1}{n\ln n} \)
45. \( \sum_{n = 1}^\infty \frac{1}{n^2+1} \)
46. \( \sum_{n = 2}^\infty \frac{1}{n(\ln n)^2} \)
47. \( \sum_{n = 1}^\infty \frac{n}{2^n} \)
48. \( \sum_{n = 1}^\infty \frac{1}{n^2+3n-5} \)
49. \( \sum_{n = 1}^\infty \frac{1}{4^n+n^2-n} \)
50. \( \sum_{n = 1}^\infty \frac{\ln n}{n} \)
51. \( \sum_{n = 1}^\infty \frac{1}{n!+n} \)
52. \( \sum_{n = 2}^\infty \frac{1}{\sqrt{n^2-1}} \)
53. \( \sum_{n = 5}^\infty \frac{1}{\sqrt{n}-2} \)
54. \( \sum_{n = 1}^\infty \frac{n^2+n+1}{n^3-5} \)
55. \( \sum_{n = 1}^\infty \frac{2^n}{5^n+10} \)
56. \( \sum_{n = 2}^\infty \frac{n}{n^2-1} \)
57. \( \sum_{n = 2}^\infty \frac{1}{n^2\ln n} \)
58. \( \sum_{n = 1}^\infty \frac{1}{n^2 -3n+5} \)
59. \( \sum_{n = 1}^\infty \frac{1}{4^n-n^2} \)
60. \( \sum_{n = 4}^\infty \frac{\ln n}{n-3} \)
61. \( \sum_{n = 1}^\infty \frac{1}{\sqrt{n^2+n}} \)
62. \( \sum_{n = 1}^\infty \frac{1}{n+\sqrt{n}} \)
63. \( \sum_{n = 1}^\infty \frac{n-10}{n^2+10n+10} \)
64. \( \sum_{n = 1}^\infty \sin\big(1/n\big) \)
65. \( \sum_{n = 1}^\infty \frac{n+5}{n^3-5} \)
66. \( \sum_{n = 1}^\infty \frac{\sqrt{n}+3}{n^2+17} \)
67. \( \sum_{n = 1}^\infty \frac{1}{\sqrt{n}+100} \)
68. \( \sum_{n = 1}^\infty \frac{n^2}{2^n} \)
69. \( \sum_{n = 1}^\infty \frac{1}{(2n+5)^3} \)
70. \( \sum_{n = 1}^\infty \frac{n!}{10^n} \)
71. \( \sum_{n = 1}^\infty \frac{\ln n}{n!} \)
72. \( \sum_{n = 1}^\infty \frac{1}{3^n+n} \)
73. \( \sum_{n = 1}^\infty \frac{n-2}{10n+5} \)
74. \( \sum_{n = 1}^\infty \frac{3^n}{n^3} \)
75. \( \sum_{n = 1}^\infty \frac{\cos (1/n)}{\sqrt{n}} \)
76. \( \sum_{n = 0}^\infty \frac{2n}{n!} \)
77. \( \sum_{n = 0}^\infty \frac{5^n-3n}{4^n} \)
78. \( \sum_{n = 0}^\infty \frac{n!10^n}{(2n)!} \)
79. \( \sum_{n = 1}^\infty \frac{1}{n} \)
80. \( \sum_{n = 1}^\infty \frac{1}{3n^3+7} \)
81. \( \sum_{n = 1}^\infty \frac{10\cdot5^n}{7^n-3} \)
82. \( \sum_{n = 1}^\infty n\cdot\left(\frac35\right)^n \)
83. \( \sum_{n = 1}^\infty \frac{2\cdot4\cdot6\cdot8\cdots 2n}{3\cdot6\cdot9\cdot12\cdots 3n} \)
84. \( \sum_{n = 1}^\infty \frac{n!}{5\cdot10\cdot15\cdots (5n)} \)
85. \( \sum_{n = 1}^\infty \frac{5^n+n^4}{7^n+n^2} \)
86. \( \sum_{n = 1}^\infty \left(\frac{2n+5}{3n+11}\right)^n \)
87. \( \sum_{n = 1}^\infty \left(\frac{.9n^2-n-3}{n^2+n+3}\right)^n \)
88. \( \sum_{n = 1}^\infty \frac{2^nn^2}{3^n} \)
89. \( \sum_{n = 1}^\infty \frac{3^n}{n^2 2^{n+1}} \)
90. \( \sum_{n = 1}^\infty \frac{1}{n^n} \)
91. \( \sum_{n = 1}^\infty \frac{4^{n+7}}{7^n} \)
92. \( \sum_{n = 1}^\infty \left(\frac{n^2-n}{n^2+n}\right)^n \)
93. \( \sum_{n = 1}^\infty \left(\frac{1}{n}-\frac{1}{n^2}\right)^n \)
94. \( \sum_{n = 1}^\infty \frac{1}{\big(\ln n\big)^n} \)
95. \( \sum_{n = 1}^\infty \frac{n^2}{\big(\ln n\big)^n} \)
96. \( \sum_{n = 1}^\infty \frac{n^2+4n-2}{n^3+4n^2-3n+7} \)
97. \( \sum_{n = 1}^\infty \frac{n^44^n}{n!} \)
98. \( \sum_{n = 1}^\infty \frac{n^2}{3^n+n} \)
99. \( \sum_{n = 1}^\infty \frac{n}{\sqrt{n^2+4n+1}} \)
100. \( \sum_{n = 1}^\infty \frac{3^n}{n^n} \)
101. \( \sum_{n = 1}^\infty \frac{(n!)^3}{(3n)!} \)
102. \( \sum_{n = 1}^\infty \frac{1}{\ln n} \)
103. \( \sum_{n = 1}^\infty \left(\frac{n+2}{n+1}\right)^n \)
104. \( \sum_{n = 2}^\infty \frac{n^3}{\big(\ln n\big)^n} \)
105. \( \sum_{n = 1}^\infty \left(\frac{1}{n}-\frac{1}{n+2}\right) \)
106. \( \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^2} \)
107. \( \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{\sqrt{n!}} \)
108. \( \sum_{n = 0}^\infty (-1)^{n}\frac{n+5}{3n-5} \)
109. \( \sum_{n = 1}^\infty (-1)^{n}\frac{2^n}{n^2} \)
110. \( \sum_{n = 0}^\infty (-1)^{n+1}\frac{3n+5}{n^2-3n+1} \)
111. \( \sum_{n = 1}^\infty \frac{(-1)^{n}}{\ln n+1} \)
112. \( \sum_{n = 2}^\infty (-1)^n\frac{n}{\ln n } \)
113. \( \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{1+3+5+\cdots+(2n-1) } \)
114. \( \sum_{n = 1}^\infty \cos(\pi n) \)
115. \( \sum_{n = 2}^\infty \frac{\sin\big((n+1/2)\pi\big)}{n\ln n} \)
116. \( \sum_{n = 0}^\infty \left(-\frac23\right)^n \)
117. \( \sum_{n = 0}^\infty (-e)^{-n} \)
118. \( \sum_{n = 0}^\infty \frac{(-1)^nn^2}{n!} \)
119. \( \sum_{n = 0}^\infty (-1)^n2^{-n^2} \)
120. \( \sum_{n = 1}^\infty \frac{(-1)^n}{\sqrt{n}} \)
121. \( \sum_{n = 1}^\infty \frac{(-1000)^n}{n!} \)

1. Given that \( \sum_{n = 1}^\infty a_n \) converges, state which of the following series converges, may converge, or does not converge.
   • \( \sum_{n = 1}^\infty \frac{a_n}n \)
   • \( \sum_{n = 1}^\infty a_na_{n+1} \)
   • \( \sum_{n = 1}^\infty (a_n)^2 \)
   • \( \sum_{n = 1}^\infty na_n \)
   • \( \sum_{n = 1}^\infty \frac{1}{a_n} \)
2. Let \( (a_n)_{n \geq 0} \) be a sequence. Is it possible for \( \sum_{n = 0}^\infty a_n \) and \( \sum_{n = 0}^\infty \frac{1}{a_n} \) to both converge? Either give an example to show that this happens for some sequence, or give a rigorous explanation to show that it cannot happen.

For each series below, complete each of the following.

• Show that the series converges.
• Use the Alternating Series Estimation Theorem to obtain an estimate for the error \( \epsilon \) in estimation using the \( n^{\text{th}} \) partial sum.
• Use the Alternating Series Estimation Theorem to obtain the smallest \( n \) such that the \( n^{\text{th}} \) partial sum has error at most \( \epsilon \).

1. \( \sum_{n = 1}^\infty \frac{(-1)^n}{\ln (n+1)}, \quad n = 5 \)
2. \( \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^4}, \quad n = 4 \)
3. \( \sum_{n = 0}^\infty \frac{(-1)^{n}}{n!}, \quad n = 6 \)
4. \( \sum_{n = 0}^\infty \left(-\frac12\right)^n, \quad n = 9 \)
5. \( \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^4}, \quad \epsilon = 0.001 \)
6. \( \sum_{n = 0}^\infty \frac{(-1)^{n}}{n!}, \quad \epsilon = 0.0001 \)
7. \( \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1}, \quad \epsilon = 0.001 \)
8. \( \sum_{n = 0}^\infty \frac{(-1)^{n}}{(2n)!}, \quad \epsilon = 10^{-8} \)

1. Compute the radius and interval of convergence of each power series below.
   • \( \sum_{n = 0}^\infty 2^nx^n \)
   • \( \sum_{n = 1}^\infty \frac{1}{n^2}x^n \)
   • \( \sum_{n = 0}^\infty \frac{1}{n!}x^n \)
   • \( \sum_{n = 0}^\infty \frac{(-1)^n}{(2n)!}x^{2n} \)
   • \( \sum_{n = 0}^\infty \frac{(-1)^{n+1}}{n!}x^{n} \)
   • \( \sum_{n = 0}^\infty nx^{n} \)
   • \( \sum_{n = 1}^\infty \frac{(-1)^n(x-3)^{n}}{n} \)
   • \( \sum_{n = 0}^\infty \frac{(x+4)^{n}}{n!} \)
   • \( \sum_{n = 0}^\infty \frac{x^{n}}{2^n} \)
   • \( \sum_{n = 0}^\infty \frac{(-1)^n(x-5)^{n}}{10^n} \)
   • \( \sum_{n = 0}^\infty 5^n(x-1)^{n} \)
   • \( \sum_{n = 0}^\infty (-2)^nx^{n} \)
   • \( \sum_{n = 0}^\infty \sqrt{n}x^{n} \)
   • \( \sum_{n = 0}^\infty \frac{n}{3^n}x^{n} \)
   • \( \sum_{n = 0}^\infty \frac{3^n}{n!}(x-5)^{n} \)
   • \( \sum_{n = 0}^\infty (-1)^nn!(x-10)^{n} \)
   • \( \sum_{n = 1}^\infty \frac{x^n}{n^2} \)
   • \( \sum_{n = 1}^\infty \frac{(x+2)^n}{n^3} \)
   • \( \sum_{n = 0}^\infty n!\left(\frac x{10}\right)^n \)
   • \( \sum_{n = 0}^\infty n^2\left(\frac{x+4}{4}\right)^n \)
   • \( \sum_{n = 0}^\infty nx^n \)
   • \( \sum_{n = 1}^\infty \frac{x^n}{n} \)
   • \( \sum_{n = 0}^\infty \left(\frac{x}{2}\right)^n \)
   • \( \sum_{n = 0}^\infty (-3x)^n \)
   • \( \sum_{n = 0}^\infty \frac{(-1)^nx^{2n}}{(2n)!} \)
   • \( \sum_{n = 0}^\infty \frac{(-1)^nx^{n}}{n!} \)
2. Solve each of the following differential equations using power series. If possible, give an equivalent expression for the solution using the table of known Taylor-Maclaurin series.
   • \( y' = 3y ,\quad y(0) = 1 \)
   • \( y' = 5y ,\quad y(0) = 5 \)
   • \( y' = y^2,\quad y(0) = 1 \)
   • \( y' = y+1,\quad y(0) = 1 \)
   • \( y'' = -y,\quad y(0) = 0, y'(0) = 1 \)
   • \( y'' = 2y,\quad y(0) = 1, y'(0) = 1 \)

Compute the Maclaurin polynomial of degree \( n \) for each function \( f \) below.

1. \( f(x) = e^{-x}, \quad n = 3 \)
2. \( f(x) = \sin x, \quad n = 8 \)
3. \( f(x) = x\cdot e^x, \quad n = 5 \)
4. \( f(x) = \tan x, \quad n = 6 \)
5. \( f(x) = e^{2x}, \quad n = 4 \)
6. \( f(x) = \frac1{1-x}, \quad n = 4 \)
7. \( f(x) = \frac1{1+x}, \quad n = 4 \)
8. \( f(x) = \frac1{1+x}, \quad n = 7 \)

Compute the Taylor polynomial of degree \( n \) centred at \( c \) for each function \( f \) below.

1. \( f(x) = \sqrt x, \quad n = 4, \quad c = 1 \)
2. \( f(x) = \ln (x+1), \quad n = 4, \quad c = 1 \)
3. \( f(x) = \cos x, \quad n = 6, \quad c = \pi/4 \)
4. \( f(x) = \sin x, \quad n = 5, \quad c = \pi/6 \)
5. \( f(x) = \frac1x, \quad n = 5, \quad c = 2 \)
6. \( f(x) = \frac{1}{x^2}, \quad n = 8, \quad c = 1 \)
7. \( f(x) = \frac{1}{x^2+1}, \quad n = 3, \quad c = -1 \)
8. \( f(x) = x^2\cos x, \quad n = 2, \quad c = \pi \)

Approximate…

1. \( \sin 0.1 \) with the Maclaurin polynomial of degree 3.
2. \( \cos 1 \) with the Maclaurin polynomial of degree 4.
3. \( \sqrt{10} \) with the Taylor polynomial of degree 2 centred at \( x = 9 \).
4. \( \ln1.5 \) with the Taylor polynomial of degree 3 centred at \( x = 1 \).

Find a formula for the \( n^{th} \) term of the Taylor series of each function \( f \) below.

1. \( f(x) = e^x \) centred at \( x = 0 \)
2. \( f(x) = \cos x \) centred at \( x = 0 \)
3. \( \ds f(x) = \frac{1}{1-x} \) centred at \( x = 0 \)
4. \( \ds f(x) = \frac{1}{1+x} \) centred at \( x = 0 \)
5. \( \ds f(x) = \ln x \) centred at \( x = 1 \)

For some function \( f(x) \), the Maclaurin polynomial of degree \( 4 \) is \( p_4(x) = 6+3x-4x^2+5x^3-7x^4 \).

1. What is \( p_2(x) \)?
2. What is \( f^{(3)}(0) \)?

Find \( n \) such that the Maclaurin polynomial of degree \( n \) for \( f \) approximates the given value within \( .0001 \) of the actual value.

1. \( f(x) = e^x \) approximating \( e \)
2. \( f(x) = \cos x \) approximating \( \cos \pi/3 \)
3. \( f(x) = \sin x \) approximating \( \cos \pi \)
4. \( f(x) = \sqrt x \), approximating \( \sqrt 3 \) (use the Taylor polynomial of degree \( n \) centred at \( x = 4 \) for this one)

Compute the Taylor series with the given centre \( c \) for \( f \).

1. \( f(x) = \sin x \);\quad \( c = 0 \)
2. \( f(x) = \cos x \);\quad \( c = \pi/2 \)
3. \( f(x) = 1/x \);\quad \( c = 1 \)
4. \( f(x) = e^x \);\quad \( c = 0 \)
5. \( f(x) = 1/(1-x) \);\quad \( c = 0 \)
6. \( f(x) = \tan^{-1}x \);\quad \( c = 0 \)
7. \( f(x) = e^{-x} \);\quad \( c = 0 \)
8. \( f(x) = \ln(1+x) \);\quad \( c = 0 \)
9. \( f(x) = x/(x+1) \);\quad \( c = 1 \)
10. \( f(x) = \sin x \);\quad \( c = \pi/4 \)

Establish the identities below via Taylor-Maclaurin Series.

1. \( \cos(-x) = \cos x \)
2. \( \sin(-x) = -\sin x \)
3. \( \frac{d}{dx}\big(\sin x\big) = \cos x \)
4. \( \frac{d}{dx}\big(\cos x\big) = -\sin x \)

Compute power series for each of the following via the table of common Taylor-Maclaurin Series.

1. \( f(x) = \cos \big(x^2\big) \)
2. \( f(x) = e^{-x} \)
3. \( f(x) = \sin\big(2x+3\big) \)
4. \( f(x) = \tan^{-1}\big(x/2\big) \)
5. \( f(x) = e^x\sin x \)\quad (only find the first 4 terms)
6. \( f(x) = (1+x)^{1/2}\cos x \)\quad (only find the first 4 terms)

Compute the integrals below using Taylor-Maclaurin Series.

1. \( \int_0^{\sqrt{\pi}} \sin \big(x^2\big)\ dx \)
2. \( \int_0^{\pi^2/4} \cos \big(\sqrt{x}\big)\ dx \)