Question 1

Evaluate the infinite series $$\sum\limits_{k=1}^{\infty} \frac{3k - 5k}{10k}. $$Does the series converge or diverge?

Accepted Answer

The series diverges because each term is $$-\frac{2}{10}$$, so the series is a divergent constant series.

Question 2

Determine whether the series $$\sum\limits_{k=1}^{\infty} 3(1.001)^k$$ converges or diverges.

Accepted Answer

The series diverges because $$|1.001| \u003e 1.$$

Question 3

Express the repeating decimal $$0.314141414\ldots as a $$ratio of two integers.

Accepted Answer

$$\frac{311}{990}$$

Question 4

Use the Divergence Test to determine whether the series $$\sum\limits_{k=1}^{\infty} \frac{2k^4$ + k^3$ + 1}{3k^4$ + 4}$$ converges or diverges.

Accepted Answer

The series diverges because the limit of the terms is $$\frac{2}{3} 
eq 0.$$

Question 5

Consider the series $$\sum\limits_{k=1}^{\infty} \frac{2k^4$ + k^3$ + 1}{3k^6$ + 4}. $$Use the Limit Comparison Test with $$\sum\limits_{k=1}^{\infty} \frac{1}{k^2$} to $$determine convergence.

Accepted Answer

The series converges because the limit comparison yields a positive finite number.

Question 6

For the series $$\sum\limits_{k=1}^{\infty} \frac{7 + \sin k}{k^2$}$$, use the Comparison Test with $$\sum\limits_{k=1}^{\infty} \frac{8}{k^2$} to $$determine convergence.

Accepted Answer

The series converges because $$\frac{7 + \sin k}{k^2$} \u003c \frac{8}{k^2$}$$ and $$\sum \frac{8}{k^2$}$$ converges.

Question 7

Use the Root Test to determine whether the series $$\sum\limits_{k=1}^{\infty} \left(\frac{k}{2k + 3}\right)^k$$ converges.

Accepted Answer

The series diverges because the limit is greater than $$1.$$

Question 8

Use the Ratio Test to determine whether the series $$\sum\limits_{k=1}^{\infty} k e$$^{-k}$$ converges.

Accepted Answer

The series converges because the ratio is $$e^{-1}$$ \u003c 1$$.

Question 9

Use the Integral Test to determine whether the series $$\sum\limits_{k=1}^{\infty} \frac{e^k}{1 + e$$^{2k}$$}$$ converges.

Accepted Answer

The series converges because the integral converges.

Question 10

Determine whether the series $$\sum\limits_{k=1}^{\infty} \frac{(-1)^k}{k$$^{5/2}$$}$$ converges absolutely, converges conditionally, or diverges.

Accepted Answer

Converges absolutely because $$\sum \frac{1}{k$$^{5/2}$$}$$ converges.

Question 11

Determine whether the series $$\sum\limits_{k=1}^{\infty} \frac{(-1$$)^{k+1}$$}{k$$^{3/7}$$}$$ converges absolutely, converges conditionally, or diverges.

Accepted Answer

Diverges because $$p \u003c 1.$$

Question 12

A pharmaceutical company is analyzing the cost of producing a drug, modeled by the series $$\sum\limits_{k=1}^{\infty} \frac{1000}{2^k}. $$What is the total cost as $$k 	o \infty$$?

Accepted Answer

$$2000$$

Question 13

Suppose a population of bacteria doubles every hour and starts at 1. What is the sum $$\sum\limits_{k=0}^{\infty} 2^k$$?

Accepted Answer

The series diverges.

Question 14

A radioactive substance decays such that the amount remaining after $$k$$ years is modeled by $$A_k = 100 \left(\frac{1}{2}\right)^k. $$What is the total amount decayed after infinite years?

Accepted Answer

$$100$$

Question 15

Which of the following series converges by the $$p-$$series test?

Accepted Answer

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

Question 16

A medical researcher models the effectiveness of a treatment by the series $$\sum\limits_{k=1}^{\infty} \frac{(-1$$)^{k+1}$$}{k}. $$What can be said about this series?

Accepted Answer

It converges conditionally.

Question 17

Which test is most appropriate to determine the convergence of $$\sum\limits_{k=1}^{\infty} \frac{1}{k \ln k}$$?

Accepted Answer

Integral Test

Question 18

A financial analyst models compound interest with the series $$\sum\limits_{k=1}^{\infty} \left(\frac{1}{1.05}\right)^k. $$Does this series converge?

Accepted Answer

Yes, because $$\left|\frac{1}{1.05}\right| \u003c 1.$$

Practice Exam 3 – Infinite Series and Convergence Tests (Calculus II)

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