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You CAN Ace Calculus

17calculus > exam list > exam 1

 infinite series all infinite series tests and topics

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

Infinite Series Exam 1

This page contains a complete infinite series exam with worked out solutions.

Exam Details

Tools

Time

1 hour

Calculators

no

Questions

13

Formula Sheet(s)

table of series tests

Total Points

100

Other Tools

none

Instructions:
- For each problem, correct answers are worth 10%. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).

Section 1

3 questions - 20 points total

Do the following sequences $$\{ {a_n} \}$$ converge or diverge as $$n \to \infty$$? If the sequence converges, find its limit. Justify your answers.

Question 1

$$a_n=2+(-1)^n$$

solution

Question 2

$$\displaystyle{ a_n = \frac{n}{e^n} }$$

solution

Question 3

$$\displaystyle{ a_n = \left( 1 + \frac{2}{n} \right)^n }$$

solution

Section 2

4 questions - 20 points total

Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.

Question 4

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^3}{3^n}} }$$

solution

Question 5

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n^2+1} } }$$

solution

Question 6

$$\displaystyle{ \sum_{n=1}^{\infty}{n \sin\left( \frac{1}{n} \right) } }$$

solution

Question 7

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

solution

Section 3

2 questions - 20 points total

Determine whether the following series converge or diverge and justify your answer.

Question 8

$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n \ln(n)} } }$$

solution

Question 9

$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{(-1)^n \ln(n)}{n} } }$$

solution

Section 4

4 questions - 40 points total

Question 10

Does this series diverge, converge conditionally or converge absolutely? Justify your answer.
$A = 1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \frac{1}{5^2} - \frac{1}{6^2} + \frac{1}{7^2} - . . .$

solution

Question 11

Let $$\displaystyle{S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + }$$ $$\displaystyle{ \frac{1}{4^2} + \frac{1}{5^2} + }$$ $$\displaystyle{\frac{1}{6^2} + \frac{1}{7^2} + . . . }$$

Given that $$S = \pi^2/6$$, find the sum A given in question 10.
(Hint: Consider $$S - A$$ and express it in terms of $$S$$.)

solution

Question 12

State the definition for a sequence $$\{ a_n \}$$ to converge to a limit $$L$$ as $$n \to \infty$$.

solution

Question 13

If $$\displaystyle{ a_n = \frac{1}{\sqrt{n}} }$$ for $$n = 1, 2, 3, . . .$$ prove from the definition that $$\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }$$.

solution

12