10.1 Flashcards
limn→∞(an+bn)
A+B
limn→∞(an−bn)
A−B
limn→∞(k⋅bn)
k⋅B
limn→∞(an⋅bn)
A⋅B
limn→∞an/bn if B does not equal 0
A/B
How would we find the limit of a sequence like cosn/n
By the sandwich theorem it approaches 0
How would we find the limit of a sequence like 1/(2^n)
By the sandwich theorem it approaches 0
limn→∞ ln(n)/n
0
limn→∞ n^(1/n)
1
limn→∞ x^(1/n) where x>0
1
limn→∞ x^n if |x|<1
0
limn→∞ (1+(x/n))^n
e^x
limn→∞ (x^n)/n!
0
What is a recursively defined sequence
One where you are given the value of an initial term and a recursion formula to calculate the rest of the terms
What is a bounded sequence
One which is bounded both from above and below
Are convergent sequences bounded
Yes
What is a monotonic sequence
One that is either non-decreasing or non-increasing
What does the monotonic sequence theorem say
If a sequence is both bounded and monotonic, it converges
Is a convergent series monotonic
Not necessarily
What are the four main strategies for finding the limit of a sequence?
-Direct evaluation of limn->inf an
-Monotonic sequence theorem (if a sequence is both bounded and monotonic, it converges)
-L’Hopital’s rule
-Squeeze theorem
How do you check if a sequence is monotonic
Either use the derivative or take a(n+1)-an
What is the limit substitution strategy and when can you apply it?
If a series is monotonic and bounded you can replace the limit with L and solve for it
