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Chapter 2 Problems

Thirty chapter problems: 19 from the original book and 11 added for more practice.

Read the chapter problems → · Chapter solutions →

Original problem

If the next term of a sequence is either always greater than or always less than the previous term, then it is known as what type of sequence?

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Worked solution to problem 1 →
Original problem

Find the common difference for the sequence
{1, -5, -11, -17, ...}

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Worked solution to problem 2 →
Original problem

Find the common ratio for the sequence

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Worked solution to problem 3 →
Original problem

For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.
{3n2-n}n=1

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Worked solution to problem 4 →
Original problem

For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.
n=1

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Worked solution to problem 5 →
Original problem

For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.
{2-ne-n}n=1

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Worked solution to problem 6 →
Original problem

For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.
n=1

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Worked solution to problem 7 →
Original problem

For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.
n=2

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Worked solution to problem 8 →
Original problem

For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.
n=1

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Worked solution to problem 9 →
Original problem

Find the 4th partial sum of the sequence
n=1

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Worked solution to problem 10 →
Original problem · clarified

Find the 5th partial sum of the sequence
n=1
Correction: the fifth partial sum adds all five terms. The equals sign before a4 in the printed answer is replaced by a plus sign.

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Original problem

For questions 12-19, determine if the series converges or diverges.

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Worked solution to problem 12 →
Original problem

For questions 12-19, determine if the series converges or diverges.
13.

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Worked solution to problem 13 →
Original problem

For questions 12-19, determine if the series converges or diverges.
14.

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Original problem · clarified

For questions 12-19, determine if the series converges or diverges.
$\sum_{n = 3}^{\infty} \frac{1}{n \ln n \ln (\ln n)}$
Clarification: this web problem begins at n=3. The source begins at n=1, where its logarithmic term is undefined; n=3 also gives a positive continuous function for the integral-test proof.

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Worked solution to problem 15 →
Original problem

For questions 12-19, determine if the series converges or diverges.

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Worked solution to problem 16 →
Original problem

For questions 12-19, determine if the series converges or diverges.

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Worked solution to problem 17 →
Original problem

For questions 12-19, determine if the series converges or diverges.

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Worked solution to problem 18 →
Original problem

For questions 12-19, determine if the series converges or diverges.

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Worked solution to problem 19 →
Added problem

An arithmetic sequence has $a_{1} = 7$ and common difference $- 4$ . Find $a_{n}$ , $a_{15}$ , and the sum of its first 15 terms.

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Worked solution to problem 20 →
Added problem

Determine whether the sequence $a_{n} = 3 (- \frac{1}{2})^{n - 1}$ , $n \geq 1$ , converges, and find its limit if it does.

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Worked solution to problem 21 →
Added problem

Find $\lim_{n \to \infty} \frac{n^{2}}{3 n^{2} + 1}$ .

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Worked solution to problem 22 →
Added problem

Find $\lim_{n \to \infty} (\sqrt{n^{2} + 3 n} - n)$ .

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Worked solution to problem 23 →
Added problem

Let $a_{1} = 1$ and $a_{n + 1} = \frac{a_{n} + 2}{2}$ . Prove that the sequence increases and is bounded, then find its limit.

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Worked solution to problem 24 →
Added problem

A series has partial sums $S_{N} = \frac{N}{N + 1}$ for $N \geq 1$ . Find its nth term and its infinite sum.

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Worked solution to problem 25 →
Added problem

Determine whether $\sum_{n = 1}^{\infty} \frac{n}{n + 1}$ converges. State the test you use.

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Worked solution to problem 26 →
Added problem

Use the integral test to determine whether $\sum_{n = 1}^{\infty} \frac{1}{n^{3 / 2}}$ converges.

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Worked solution to problem 27 →
Added problem

Determine whether $\sum_{n = 2}^{\infty} \frac{1}{n \ln n}$ converges using the integral test.

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Added problem

Use the integral test to determine whether $\sum_{n = 2}^{\infty} \frac{\ln n}{n^{2}}$ converges. Verify its hypotheses.

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Worked solution to problem 29 →
Added problem

Find the partial sum formula and infinite sum of $\sum_{n = 1}^{\infty} \frac{1}{n^{2} + 4 n + 3}$ .

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Worked solution to problem 30 →

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