Q: Which of the following statements is TRUE regarding the convergence of a series $$\sum\limits_{n=1}^{\infty} a_n$$?

If $$\lim\limits_{n \to \infty} a_n \neq 0$$, then the series diverges.

Q: Determine if the series $$\sum\limits_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$ n}{n+1}$$ converges or diverges.

It diverges because $$\lim\limits_{n \to \infty} \frac{n}{n+1} = 1 \neq 0.$$

Q: Which of the following is the correct test for divergence of a series $$\sum a_n$$?

If $$\lim\limits_{n \to \infty} a_n$$ does not exist or $$\lim\limits_{n \to \infty} a_n \neq 0$$, then the series diverges.

Question 1

Which of the following best defines an infinite series?

Accepted Answer

An infinite sum of the terms of a sequence, written as $$\sum\limits_{n=1}^{\infty} a_n$$

Question 2

Given the geometric series $$5 - \frac{10}{3} + \frac{20}{9} - \cdots$$, what is its sum?

Accepted Answer

$$3$$

Question 3

Which of the following is the correct formula for the sum of an infinite geometric series with first term $$a_1$$ and common ratio $$r ($$where $$|r| \u003c 1$$)?

Accepted Answer

$$S = \frac{a_1$}{1 - r}$$

Question 4

Evaluate the sum $$\sum\limits_{n=1}^{\infty} \frac{1}{n(n+1)}.$$

Accepted Answer

$$1$$

Question 5

Which of the following statements is TRUE regarding the convergence of a series $$\sum\limits_{n=1}^{\infty} a_n$$?

Accepted Answer

If $$\lim\limits_{n 	o \infty} a_n 
eq 0$$, then the series diverges.

Question 6

Determine if the series $$\sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n}}$$ converges or diverges.

Accepted Answer

It diverges because it is a $$p-$$series with $$p = \frac{1}{2} \u003c 1.$$

Question 7

Find the sum of the geometric series $$\sum\limits_{n=4}^{\infty} 7 \left(-\frac{3}{4}\$$right)^{n-1}$$.

Accepted Answer

$$-\frac{189}{64}$$

Question 8

Which of the following is the infinite series representation for $$\sin(1) as $$given in the materials?

Accepted Answer

$$\sin(1) = 1 - \frac{1}{3!} + \frac{1}{5!} - \frac{1}{7!} + \frac{1}{9!} - \frac{1}{11!} + \cdots$$

Question 9

Evaluate the sum $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots ($$Zeno's Paradox).

Accepted Answer

$$1$$

Question 10

Determine if the series $$\sum\limits_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$ n}{n+1}$$ converges or diverges.

Accepted Answer

It diverges because $$\lim\limits_{n 	o \infty} \frac{n}{n+1} = 1 
eq 0.$$

Question 11

Find the sum of the series $$\sum\limits_{n=1}^{\infty} \left( \frac{1}{n(n+1)} + \frac{1}{2^n} \right).$$

Accepted Answer

$$2$$

Question 12

Which of the following is NOT a property of convergent series?

Accepted Answer

If $$\sum a_n$$ and $$\sum b_n$$ are convergent, then $$\sum (a_n \cdot b_n) is $$convergent.

Question 13

Find the sum of the geometric series $$\sum\limits_{n=1}^{\infty} 2$$^{n}$$ 3$$^{1-n}$$.

Accepted Answer

$$6$$

Question 14

Which of the following is the correct test for divergence of a series $$\sum a_n$$?

Accepted Answer

If $$\lim\limits_{n 	o \infty} a_n$$ does not exist or $$\lim\limits_{n 	o \infty} a_n 
eq 0$$, then the series diverges.

Question 15

A pharmaceutical company is testing a new drug and records the amount of the drug in the bloodstream after each hour as a geometric sequence with first term $$a_1 = 100$ mg and common ratio r = 0.8$. What is the total amount of drug present in the bloodstream after an infinite number of hours?

Accepted Answer

$$500 mg$$

Question 16

Which of the following series converges?

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \frac{1}{2^n}$$

11.2 Series: Infinite Series, Geometric Series, and Convergence

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