Q: Does the series $$\sum_{n=1}^{\infty} \frac{n+5}{2n+3}$$ converge or diverge?

It diverges because $$\lim_{n \to \infty} \frac{n+5}{2n+3} = \frac{1}{2} \neq 0$$

Q: If a series $$\sum_{n=1}^{\infty} a_n$$ converges, which of the following must be true?

$$\lim_{n \to \infty} a_n = 0$$

Question 1

Which of the following is the correct partial fraction decomposition for $$\frac{2}{n^2$ + 4n + 3}$$?

Accepted Answer

$$\frac{2}{(n+1)(n+3)} = \frac{1}{n+1} - \frac{1}{n+3}$$

Question 2

Consider the series $$\sum_{n=1}^{\infty} \frac{2}{n^2$ + 4n + 3}. $$What is the value of its sum?

Accepted Answer

$$\frac{3}{2}$$

Question 3

Which test is most appropriate to determine the convergence of the series $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$$?

Accepted Answer

The p-series test

Question 4

Does the series $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$$ converge or diverge?

Accepted Answer

It diverges because $$p = \frac{1}{2} \u003c 1 in $$the p-series test.

Question 5

Given the geometric series $$\sum_{n=1}^{\infty} 6 \left( \frac{2}{3} \$$right)^{n-1}$$, what is the sum?

Accepted Answer

$$18$$

Question 6

For the series $$\sum_{n=1}^{\infty} \frac{n+5}{2n+3}$$, which test would best determine its convergence?

Accepted Answer

The divergence (nth-term) test

Question 7

Does the series $$\sum_{n=1}^{\infty} \frac{n+5}{2n+3}$$ converge or diverge?

Accepted Answer

It diverges because $$\lim_{n 	o \infty} \frac{n+5}{2n+3} = \frac{1}{2} 
eq 0$$

Question 8

What is the sum of the geometric series $$\sum_{n=1}^{\infty} 7(2$$^{n-1}$$)/(-5)^n$$?

Accepted Answer

$$-2$$

Question 9

Convert the repeating decimal $$0.\overline{11} to a $$rational number.

Accepted Answer

$$\frac{1}{9}$$

Question 10

Given the series $$\sum_{n=1}^{\infty} \left( \frac{1}{n^5$} + \frac{4}{n^3$ - 1} \right)$$, which part determines the convergence?

Accepted Answer

Both terms must be considered, but $$\frac{1}{n^5$}$$ converges and $$\frac{4}{n^3$ - 1}$$ also converges by comparison to $$\frac{4}{n^3$}.$$

Question 11

What is the sum of the series $$3 - 6/5 + 12/25 - 24/125 + 48/625 - \ldots$$?

Accepted Answer

$$1$$

Question 12

Which of the following is the correct formula for the sum of an infinite geometric series with first term $$a$$ and common ratio $$r$$ ($$|r| \u003c 1$$)?

Accepted Answer

$$S = \frac{a}{1 - r}$$

Question 13

If a series $$\sum_{n=1}^{\infty} a_n$$ converges, which of the following must be true?

Accepted Answer

$$\lim_{n 	o \infty} a_n = 0$$

Question 14

Which of the following series is a telescoping series?

Accepted Answer

$$\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right)$$

Question 15

Which of the following is a necessary condition for a geometric series $$\sum_{n=0}^{\infty} ar^n to $$converge?

Accepted Answer

$$|r| \u003c 1$$

Question 16

Which of the following series diverges?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

Question 17

A chemical reaction produces a sequence of concentrations modeled by $$C_n = 10 \left( \frac{1}{2} ight)^n. $$What is the limiting concentration as $$n 	o \infty$$?

Accepted Answer

$$0$$

Question 18

A patient receives a dose of medication every hour, with each dose being half the previous. If the first dose is 100 mg, what is the total amount received after infinite hours?

Accepted Answer

$$200 mg$$

Question 19

Which of the following is the correct sum for the series $$\sum_{n=1}^{\infty} \frac{1}{2^n}$$?

Accepted Answer

$$1$$

Question 20

If $$\sum_{n=1}^{\infty} a_n is a $$convergent series, which of the following statements is always true?

Accepted Answer

The sequence $$a_n$$ approaches zero as $$n$$ increases.

Calculus II: Series and Convergence Study Notes (Chapter 11)

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