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

1)Estimate the limited of

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

Practice:

2)

3)

4)

Not every problem has a limit…

5)

6)

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 +…

Add the following:

4 + .001 + .0001 + .00001 + .000001

Sum of Infinite Geometric Series: (assuming convergent)

7)Find the sum of the series: 21 - 3 + 3/7-…

8) Find the sum of (6/5) + (4/5) + (8/15)+…