Calculus Exam 3 Material

Question 1

What is the formula for the sum of an infinite geometric series?

Answer 1

a1 / (1 - r); Where r is the common ratio, and the absolute value of r is less than 1

Question 2

What is the sum of the following infinite geometric series?
3 + 2 + 4/3 + 8/9 + 16/27 + …

Answer 2

3 / (1 - 2/3) = 9; r = 2/3

Question 3

Does the series 2n / (3n - 4) Converge or Diverge?

Answer 3

Diverges based on the Divergence Test; The limit as n approaches infinity does not equal 0.

Question 4

Which series has this formula, starting at the first term to infinity?
ar^n

Answer 4

Infinite geometric series

Question 5

If the absolute value of r is greater than 1 in a geometric series, then it ______.

Answer 5

diverges

Question 6

The integral test states…

Answer 6

If the integral of a positive series f(x) that decreases from 1 to infinity, converges, then the series also converges. If it diverges, then the series also diverges.

Question 7

What is a series whose sum will continually increase and always diverge known as?

Answer 7

Harmonic series

Question 8

What is the term for the following series?
(1/n) - (1/(n+1))

Answer 8

Telescoping series; first and last terms will always be the sum.

Question 9

Given the p-series 1/n^p, if p is > 1, the series will ______.
If it is < or = 1, than the series will ______.

Answer 9

Converge; diverge

Question 10

Assuming the 1st term in the series is 1 and goes to infinity, will n^-pi converge or diverge?

Answer 10

Converge

Question 11

What does the direct comparison test state?

Answer 11

Given a series > or = another series > 0…
If the larger series converges, the smaller series also converges.
If the smaller series diverges, the larger series also diverges.
If the larger series diverges initially or the smaller series converges initially, nothing can be determined.

Question 12

Given the series 1/(4+3^n), if the constant is removed, the larger series is 1/3^n. Does this series converge or diverge? Can the smaller series be determined from this answer?

Answer 12

Geometric series where |r| = 1/3. Therefore, the series converges, so 1/(4+3^n) converges as well based on the direct comparison test.

Question 13

Given the series 1/(n^3+5), if the constant is removed, the larger series is 1/n^3. Does this series converge or diverge? Can the smaller series be determined from this answer?

Answer 13

P-series where p = 3. Therefore, the series converges, so 1/(n^3+5) converges as well based on the direct comparison test.

Question 14

Given the series (n+5)/n^2, if the constant 5 is removed, the smaller series is n/n^2. Does this series converge or diverge? Can the smaller series be determined from this answer?

Answer 14

P-series (and harmonic series), where p = 1. Therefore, the series diverges, so (n+5)/n^2 diverges as well based on the direct comparison test.