12-3 infinite sequences and series. hints to solve limits: 1)rewrite fraction as sum of multiple...
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12-3 Infinite Sequences and Series
• Infinite sequence- a sequence that can infinitely many terms.
• Limit: The number in which the sequence (f(x)) gets closer and closer to.
L=
For example, what is the limit for f(x)= 1/n?= 0
Hints to solve limits:1) Rewrite fraction as sum of multiple fractions
Hint: anytime you have a number on top, and variable on bottom, the limit is 0. (ex 1/x2, 1/x3, 4/5x2)
Anytime you have a variable on top (2x, 3x, x/2) the limit is ∞, or Does Not Exist
Want a short cut?
• If degree of numerator = degree of denominator,Then the limit is the ratio of coefficientsEx)
• If degree of numerator < degree of denominator,Then the limit is 0Ex)
• If degree of numerator > degree of denominator,Then the limit does not exist
Convergent Series: An infinite series that approaches a limit
Divergent Series: An infinite series that does not approach a limit.
All arithmetic series are divergent
In a geometric series, if lrl > 1, the series is divergentif lrl <1, the series is convergent
Are the following series convergent or divergent?
1)
2)2 + 4 + 6 + 8 + 10+…
3) 10 + 8.5 + 7 + 5.5+…
4) 5, 25, 125 +…