Homework Problems

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Last update: Wednesday, 19 October 2022.

Here are problems to turn in, where the corresponding due date for each homework set is printed at the top of that homework set's section. Late homework will not be accepted. Feel free to collaborate, as long as you do not copy/give solutions from/to anyone. Remember: attempting a homework will never hurt you in my class, even if your answer is wrong.

For Friday, 21 October 2022 complete each of the indicated integrals.

1. \( \defint{0}{3}{\ln(x^2 + 1)} \)
2. \( \indef{\frac{1}{x^2\sqrt{x^2 + 1}}} \)
3. \( \indef{\frac{\sqrt{x^2 - 16}}{x^2}} \)
4. \( \indef{\sqrt{4 - x^2}} \)
5. \( \indef{\frac{1}{x(x - 1)}} \)
6. \( \indef{\sec^4(x)} \)

For Monday, 24 October 2022 complete each of the indicated integrals.

1. \( \defint{\tfrac{\pi}{6}}{\tfrac{5\pi}{3}}{\cos^3(x)\sin^{11}(x)} \)
2. \( \indef{\frac{1}{\sqrt{4x^2 - 49}}} \)
3. \( \indef{\frac{x^2}{\sqrt{1 - 9x^2}}} \)
4. \( \indef{\frac{1}{\sqrt{x^6 - x^4}}} \)
5. \( \indef{\frac{x}{x^2 - 5x + 6}} \)
6. \( \indef{x^4\arctan{x}} \)
7. \( \defint{2}{5}{\frac{1}{\sqrt{x^2 - 1}}} \)
8. \( \indef{\frac{3x^4 + 1}{x^4 + 4x^2}} \)
9. \( \defint{-3}{3}{\frac{1}{x^2 + 2x + 8}} \)

For Wednesday, 26 October 2022 complete each of the indicated integrals.

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

For Wednesday, 2 November 2022, compute the limit of each sequence indicated below, or show it does not exist.

1. \( a_n = 3^n7^{-n} \)
2. \( a_n = \sqrt{\frac{1+4n^2}{1+n^2}} \)
3. \( a_n = \frac{n^3+5}{5n^3+4n} \)
4. \( a_n = e^{-\frac{1}{\sqrt{n}}} \)

As an added (optional) challenge, repeat the above exercise for the following sequences.

1. \( a_n = \sin(n) \)
2. \( a_n = \frac{n^2}{\sqrt{n^3+4n}} \)
3. \( a_n = \sqrt[n]{n} \)

For Monday, 7 November 2022, consider the following sequences.

• \( a_n = \frac{4^n}{(n+1)!} \)
• \( b_n = (-1)^n n \)
• \( c_n = \frac{(-1)^n n}{n^2 + 1} \)
• \( d_n = \left(-\frac{1}{2}\right)^n \)
• \( e_n = \sin(3/n)\left(1+\frac{2}{n}\right)^n \)
• \( f_n = \frac{n}{n+2} \)

For each of the sequences above, complete the following; give complete justifications…

1. Compute the first four terms of the sequence.
2. Decide if the sequence is monotone increasing, monotone decreasing, or neither.
3. Decide if the sequence is bounded above, bounded below, bounded, or none of these.
4. Decide if the sequence converges or diverges. If it converges, compute the limit.

As an (optional) challenge problem…

• Define a sequence \( (g_n)_{n \geq 0} \) by \( g_0 = 1 \) and \( g_{n+1} = \frac{1}{1 + g_n} \) for all \( n \geq 0 \). Show that \( (g_n)_{n \geq 1} \) converges, and compute the limit.

For Monday, 14 November 2022, consider the following series.

1. \( \sum_{n = 0}^\infty \frac{4^n}{3^n} \)
2. \( \sum_{n = 0}^\infty \frac{1}{n \ln(n)} \)
3. \( \sum_{n = 1}^\infty \frac{1}{n^2 + 2n} \)
4. \( \sum_{n = 0}^\infty \frac{(-1)^n}{n^5} \)
5. \( \sum_{n = 0}^\infty \frac{1}{n^2 + 4} \)
6. \( \sum_{n = 0}^\infty (-1)^n 3^n 5^{-n} \)
7. \( \sum_{n = 0}^\infty \frac{5}{\sqrt{n}} \)
8. \( \sum_{n = 0}^\infty \frac{1}{\arctan(n)} \)
9. \( \sum_{n = 5}^\infty \frac{\sin(n)}{n^3 - n} \)
10. \( \sum_{n = 0}^\infty \frac{n}{16n^4 + 5} \)

Complete the following for each series above.

a

Classify the type of series (geometric, \( p \)-series, telescoping, alternating, or none of these). Note that a given series may have multiple types!

b

Determine if the series is convergent or divergent. Give a complete justification using the convergence tests we know, or by making a direct computation.

c

If geometric or telescoping, compute the sum of the series. You don't need to do this for other types of series!

As an added challenge, repeat the above exercises for the following.

Challenge 1

\( \sum_{n = 1}^\infty \frac{(-1)^n}{n^2 + n} \)

Challenge 2

\( \sum_{n = 1}^\infty \frac{1}{n^2 + kn} \) where \( k > 0 \) is an unknown positive integer.

Challenge 3

\( \sum_{n = 1}^\infty \frac{\ln(n)}{n^k} \) where \( k > 0 \) is an unknown positive real number.

Given that \(\sum_{n=1}^\infty a_n\) converges, state which of the following series converges, may converge, or does not converge.

1. \(\sum_{n=1}^\infty \frac{a_n}n\)
2. \(\sum_{n=1}^\infty a_na_{n+1}\)
3. \(\sum_{n=1}^\infty (a_n)^2\)
4. \(\sum_{n=1}^\infty na_n\)
5. \(\sum_{n=1}^\infty \frac{1}{a_n}\)

For Wednesday, 30 November 2022, complete the following exercises.

1. For each series below, does the series converge or diverge? If it converges, does it converge absolutely or conditionally? Give full justifications (including citing all of the tests you apply)!
   1. \( \sum_{n=0}^\infty \frac{(-1)^n}{n!} \)
   2. \( \sum_{n=2}^\infty \frac{(-1)^n}{\ln(n)} \)
   3. \( \sum_{n=2}^\infty \frac{1}{n\ln n} \)
   4. \( \sum_{n=0}^\infty \frac{(-1)^n}{n^2-n-1} \)
   5. \( \sum_{n=0}^\infty \frac{1}{n!+n} \)
   6. \( \sum_{n=1}^\infty \frac{(-1)^n}{\arctan(n)} \)
   7. \( \sum_{n=1}^\infty \frac{(-1)^n}{n\arctan(n)} \)
   8. \( \sum_{n=1}^\infty \frac{(-1)^n\sin(2^n)}{n^2\arctan(n)} \)
   9. \( \sum_{n=0}^\infty (\cos(2^{-n})-1) \)
2. Read about the Root Test and the Ratio Test in Apex Calculus and Strang.
   1. Apply the Root Test or the Ratio Test to each of the following series. What can you conclude?
      1. \( \sum_{n=1}^\infty \frac{(-1)^n 3^{2n}}{n^n} \)
      2. \( \sum_{n=1}^\infty \frac{n^5}{7^n} \)
      3. \( \sum_{n=0}^\infty \frac{(-1)^n n!}{1000^n} \)
      4. \( \sum_{n=1}^\infty \frac{(-1)^n n!}{n^{1000}} \)
      5. \( \sum_{n=1}^\infty \frac{(-1)^n n!}{n^n} \) (this one is a bit tricky)
   2. For what values of \( x \) do the following series converge?
      1. \( \sum_{n=0}^\infty \frac{x^n}{n!} \)
      2. \( \sum_{n=0}^\infty \frac{x^n}{7^n} \)
      3. \( \sum_{n=0}^\infty \frac{(-1)^nx^n}{n^n} \)