Series

Question 1

What does Σ mean?

Answer 1

“Sigma”
Denotes a series
Means sum of

Question 2

What are known series used for

Answer 2

To predict the nature of unfamiliar series

Question 3

What are the 3 main series

Answer 3

Harmonic Series
P-Series
Geometric Series

Question 4

What is the harmonic series and nature

Answer 4

1 / n
Diverges

Question 5

What is the p-series and its nature

Answer 5

1 / n^p

If p > 1, converges
(Converges, colossal)

If p < 1, diverges

Question 6

What is the geometric series and nature

Answer 6

ar^n

If 0 =< r < 1, the series converges

If |r| > 1, the series diverges
(R2D2)

Question 7

What is the sum of the geometric series

Answer 7

a / ( 1 - r )

a is the first term of the series
r is the number that is raised to the power

Question 8

What are the 3 theorems of convergence and divergence of non-negative series?

Answer 8

1. If an entire series converges or diverges, any part of that series will do the same
2. Multiplying a series by a constant does not change its nature
3. Adding 2 series of the same nature results in the same nature

Question 9

What are the tests for convergence or divergence of a series

Answer 9

1. Nth test
2. Geometric Test
3. Integral Test
4. P-series Test
5. Comparison Test (convergence and divergence)
6. Limit Comparison Test
7. Limit Ratio Test

Question 10

What is the nth test

Answer 10

Find limit as n —> ∞

If limit exists and is not 0, it diverges

If limit is 0, it is inconclusive. Do another test

If limit does not exist, it is inconclusive. Do another test

Question 11

What is the geometric test

Answer 11

Σ ar^n

If |r| < 1, it converges
If |r| > 1, it diverges
(R2D2)

Question 12

What is the integral test

Answer 12

b .
Σ an = lim ∫ f(x) dx = Limit
1 .
(As b approaches infinity)

If a limit exists, it converges

If a limit DOES NOT exist, it diverges

DED
Doesn’t Exist? Diverges

Question 13

What are the conditions of the integral test

Answer 13

1. Must be a non-negative series
2. f(x) must be continuous
3. Terms must decrease
4. f(n) = an
   (Term = value of the function at said term)

Question 14

What is the p-series test

Answer 14

Σ 1 / n^p

If p = 1 or |p| < 1, it Diverges
If |p| > 1, it Converges

If p is colossal it converges

Question 15

What is the comparison test for convergence

Answer 15

1. Find a similar known series “u” for series “a”
2. Compare nth terms.
3. Analyze.
   If U converges and nth term of U is larger than A, both A and U converge
   .
   If nth term of U is smaller, test is inconclusive
   .
   If U diverges, test is inconclusive

Question 16

What is the comparison test for divergence

Answer 16

1. Find a similar known series “u” for series “a”
2. Compare nth terms.
3. Analyze.
   If U diverges and nth term of U is smaller than A, both A and U diverge
   .
   If nth term of U is larger, test is inconclusive
   .
   If U converges, test is inconclusive

Question 17

What is the limit comparison test

Answer 17

Find lim an / bn
n –>∞
(Where an is the wanted series and bn is a known series)

If a non-zero limit exists, series A and B will have the same nature

If the L = 0, and bn converges, so will an
(If bn diverges, test is inconclusive)

If the L = ∞, and bn diverges, so will an
(If bn converges, test is inconclusive)

• because bn is larger and converges, limit will be 0, and an will converge
• because bn is smaller and diverges, limit will be ∞, and an will diverge

Question 18

What is the limit ratio test

Answer 18

Find lim |a(n+1) / an| = L
(As n —> ∞)

If L = 1, use another test

If L < 1, it converges
(Because the next term is smaller)

If L > 1, it diverges
(Because the next term is always larger)