Unit 10 : Infinite Sequinces And Series Flashcards
What is an infinite sequence?
An infinite sequence is a list of numbers that continues indefinitely without terminating.
True or False: A series is the sum of the terms of a sequence.
True
Fill in the blank: The _______ test is used to determine the convergence or divergence of a series by comparing it to a known series.
comparison
What does it mean for a series to converge?
A series converges if the sum of its terms approaches a finite limit as more terms are added.
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 + …
A) 1 + 1/2 + 1/4 + 1/8 + …
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?
Geometric series
Fill in the blank: A harmonic series is a specific type of ________ series.
p-series
What is the general form of a telescoping series?
A series that can be expressed as a difference of two consecutive terms.
Identify the type of series: Σ (1/n^p) where p = 2.
p-series
True or False: An alternating series is one in which the terms change sign.
True
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
B) Terms decrease to zero
What is the convergence criterion for the harmonic series?
The harmonic series diverges.
Provide an example of a telescoping series.
Σ (1/n - 1/(n+1))
In an alternating series, what test can be used to determine convergence?
The Alternating Series Test
Geometric series form?
(n=0)Σar^n or (n=1)Σar^(n-1) or (n=1)Σar^(n+1)
(c is a constraint, Σ a(n)=A, Σb(n)=B ) Σc a(n) .This expression can be rewritten as?
cA
(Σ a(n)=A, Σb(n)=B ). Σ(a(n) +/- b(n) ) = ????
A +/- B
What strat. do u use for telescoping series?
Partial fractions
What does the nth term test prove
Divergence
Nth term test summary
If the limit as n -> infinity of a(n) doesn’t =0, the series diverges
If the nth term test comes back to 0, what conclusion can be drawn?
May or may not converge
Geometric test criteria
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
Geometric series summary
If |r| is grater than or equal to (>=) 1, the series diverges. If 0<|r|<1, then the series converges
Integral test criteria
1) f must be continuous (f(n) =a(n))
2) possitive
3) decreasing in magnitude (a (n+1) < a(n) )
4) [1,inft.)
Integral test summary
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
P-series form
(n=1)(infinity)Σ(1/(n^p))
P-series test
If p>1, the series converges. If 0<p<=1 the series converges.
Direct comparison test criteria
(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)
Dirrect comparison test summary (Let 0 < s(n) ≤ b(n) for all n)
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
Limit comparison test criteria
(n=1)(inf.)Σ a(n) and (n=1)(inf.)Σ b(n) are both series with positive terms.
Limit comparison test summary
If the limit as n approaches inf. of a(n)/b(n) is =L, such that 0<L<infinity, then they both either converge or diverge
Alternating series
A series whose terms alternate from positive to negative or negative to positive
Alternating series test summary
-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)
Alt. Series test error bound
- 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.
Conditional conevergence
If (n=1)(inf.)Σ a(n) converges but (n=1)(inf.)Σ |a(n)| diverges
If (n=1)(inf.)Σ a(n) converges but (n=1)(inf.)Σ |a(n)| diverges
Conditional convergence
If (n=1)(inf.)Σ a(n) converges and (n=1)(inf.)Σ |a(n)| converges too
Absolute convergence
Geometric series sum formula (If (n=1)(inf.)Σ a(n), find the sum of the nth term)
S(n) = a(1) * (1-r^(n))/(1-r))
Ratio test
Lim (as n -> inf. of) |(a(n+1))/(a(n))| = ____
1) <1 : Converges
2) >1 and =(inf.) : Diverges
3)=1 : Nonconclusive
Root test
Lim as n approaches inf. of the nth root of |a(n)|=____
1)<1 : converges
2)>1 or inf. : Diverges
3)=1 : inconclusive
