Unit 10 : Infinite Sequinces And Series

Question 1

What is an infinite sequence?

Answer 1

An infinite sequence is a list of numbers that continues indefinitely without terminating.

Question 2

True or False: A series is the sum of the terms of a sequence.

Answer 2

True

Question 3

Fill in the blank: The _______ test is used to determine the convergence or divergence of a series by comparing it to a known series.

Answer 3

comparison

Question 4

What does it mean for a series to converge?

Answer 4

A series converges if the sum of its terms approaches a finite limit as more terms are added.

Question 5

Which of the following series is an example of a geometric series: A) 1 + 1/2 + 1/4 + 1/8 + … B) 1 + 2 + 3 + 4 + …

Answer 5

A) 1 + 1/2 + 1/4 + 1/8 + …

Question 6

What type of series is defined as the sum of the terms of the form a * r^n where a is a constant and r is the common ratio?

Answer 6

Geometric series

Question 7

Fill in the blank: A harmonic series is a specific type of ________ series.

Answer 7

p-series

Question 8

What is the general form of a telescoping series?

Answer 8

A series that can be expressed as a difference of two consecutive terms.

Question 9

Identify the type of series: Σ (1/n^p) where p = 2.

Answer 9

p-series

Question 10

True or False: An alternating series is one in which the terms change sign.

Answer 10

True

Question 11

Multiple Choice: Which of the following is NOT a characteristic of a geometric series? A) Common ratio B) Terms decrease to zero C) Converges if |r| < 1 D) Sum formula exists

Answer 11

B) Terms decrease to zero

Question 12

What is the convergence criterion for the harmonic series?

Answer 12

The harmonic series diverges.

Question 13

Provide an example of a telescoping series.

Answer 13

Σ (1/n - 1/(n+1))

Question 14

In an alternating series, what test can be used to determine convergence?

Answer 14

The Alternating Series Test

Question 15

Geometric series form?

Answer 15

(n=0)Σar^n or (n=1)Σar^(n-1) or (n=1)Σar^(n+1)

Question 16

(c is a constraint, Σ a(n)=A, Σb(n)=B ) Σc a(n) .This expression can be rewritten as?

Answer 16

cA

Question 17

(Σ a(n)=A, Σb(n)=B ). Σ(a(n) +/- b(n) ) = ????

Answer 17

A +/- B

Question 18

What strat. do u use for telescoping series?

Answer 18

Partial fractions

Question 19

What does the nth term test prove

Answer 19

Divergence

Question 20

Nth term test summary

Answer 20

If the limit as n -> infinity of a(n) doesn’t =0, the series diverges

Question 21

If the nth term test comes back to 0, what conclusion can be drawn?

Answer 21

May or may not converge

Question 22

Geometric test criteria

Answer 22

Must be a geometric series in form (n=0)Σar^n or (n=1)Σar^(n-1) or (n=1)Σar^(n+1), a not=0 , where a is the first term and r is the common ratio

Question 23

Geometric series summary

Answer 23

If |r| is grater than or equal to (>=) 1, the series diverges. If 0<|r|<1, then the series converges

Question 24

Integral test criteria

Answer 24

1) f must be continuous (f(n) =a(n))
2) possitive
3) decreasing in magnitude (a (n+1) < a(n) )
4) [1,inft.)

Question 25

Integral test summary

Answer 25

If the antiderivative of f(x) (f(n)=a(n)) from 1 to inft. converges/diverges, Σ a(n) from 1 to infinity converges/diverges

Question 26

P-series form

Answer 26

(n=1)(infinity)Σ(1/(n^p))

Question 27

P-series test

Answer 27

If p>1, the series converges. If 0

Question 28

Direct comparison test criteria

Answer 28

(n=1)(inf.)Σ s(n) (small) and (n=1)(inf.)Σb(n) (big) are series w/t positive terms. Let 0 < s(n) ≤ b(n) for all n (Must state b(n) or s(n) is greater or less than on paper)

Question 29

Dirrect comparison test summary (Let 0 < s(n) ≤ b(n) for all n)

Answer 29

1) If (n=1)(inf.)Σ b(n) (big) converges, then (n=1)(inf.)Σ s(n) (small) converges too 2) If (n=1)(inf.)Σ s(n) (small) diverges, then (n=1)(inf.)Σ b(n) (big) diverges too

Question 30

Limit comparison test criteria

Answer 30

(n=1)(inf.)Σ a(n) and (n=1)(inf.)Σ b(n) are both series with positive terms.

Question 31

Limit comparison test summary

Answer 31

If the limit as n approaches inf. of a(n)/b(n) is =L, such that 0

Question 32

Alternating series

Answer 32

A series whose terms alternate from positive to negative or negative to positive

Question 33

Alternating series test summary

Answer 33

-Alternating series (n=1)(inf.)Σ (-1)^(n) * a(n) and (n=1)(inf.)Σ (-1)^(n+1)* a(n) converge if … 1) a(n) is positive for every n (a(n)>0) 2) the limit as n approaches inf. of a(n) equals 0 3) a(n+1) < a(n)

Question 34

Alt. Series test error bound

Answer 34

- Occurs when alt series satisfies a(n+1) ≤ a(n) - The alternating series error bound says the error in your sum is at most the next term you didn't add. -In other words the error is given by the absolute value of the remainder (next term) - This helps you know how close your answer is to the actual sum.

Question 35

Conditional conevergence

Answer 35

If (n=1)(inf.)Σ a(n) converges but (n=1)(inf.)Σ |a(n)| diverges

Question 36

If (n=1)(inf.)Σ a(n) converges but (n=1)(inf.)Σ |a(n)| diverges

Answer 36

Conditional convergence

Question 37

If (n=1)(inf.)Σ a(n) converges and (n=1)(inf.)Σ |a(n)| converges too

Answer 37

Absolute convergence

Question 38

Geometric series sum formula (If (n=1)(inf.)Σ a(n), find the sum of the nth term)

Answer 38

S(n) = a(1) * (1-r^(n))/(1-r))

Question 39

Ratio test

Answer 39

Lim (as n -> inf. of) |(a(n+1))/(a(n))| = ____ 1) <1 : Converges 2) >1 and =(inf.) : Diverges 3)=1 : Nonconclusive

Question 40

Root test

Answer 40

Lim as n approaches inf. of the nth root of |a(n)|=____ 1)<1 : converges 2)>1 or inf. : Diverges 3)=1 : inconclusive