MATH 1132Q Lecture Notes - Lecture 8: Monotonic Function
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MATH 1132Q , Lecture 8 , Section 11.1 : Sequences and Series
- An important question we want to answer is whether we can find a good
polynomial approximation to any function.
Big Question : Can we make sense of a sum of infinitely many numbers?
Sequence - an ordered list of numbers a , a2 , a3 , … (continues to infinity). Each
number is a term in the sequence.
Example:
a) Find the next two terms in the sequence : 1 , 2 , 3 , 4 , 5 , 6 , 7 .
b) Find a general formula for an , the nth term in the sequence
→ an = n or {an = n}n=1 to ∞
c) Find the limit of this sequence (similar to finding the limit of a function)
→ lim(n→∞) an = lim(n→∞) n = ∞ → DNE (sequence diverges)
Note
: If the limit exists, we say the sequence converges. Otherwise, the sequence diverges.
Example:
a) Find the next two terms in the sequence : ½ , ⅔ , ¾ , ⅘ , ⅚ , 6/7 , ⅞ .
b) Find a general formula for an , the nth term in the sequence
→ For an , numerator = n , denominator = n+1
→ {an = n/n+1}n=1 to ∞
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Document Summary
Math 1132q , lecture 8 , section 11. 1 : sequences and series. An important question we want to answer is whether we can find a good polynomial approximation to any function. Sequence - an ordered list of numbers a , a 2 , a 3 , (continues to infinity). Each number is a term in the sequence. Lim (n ) a n = lim (n ) n = dne (sequence diverges) Note : if the limit exists, we say the sequence converges . For a n , numerator = n , denominator = n+1. C) find the limit of this sequence (similar to finding the limit of a function) = lim (n ) (n/n) / [ (n/n) + (1/n) ] Dne , since i 7/5 i > 1. Note : observe the numerator and denominator have the same degree, limit is fraction of coefficient of highest power of n in the numerator and denominator.
