Calculus
Lesson
49
Series
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An Infinite Series is defined as
A Partial Sum is 
So 
and so on.
Definition of convergence of a series:
If the SEQUENCE of partial sums converges, then the SERIES converges (to S)
Example: 
but 
The next lessons will all either deal with:
1) finding the Sum of a series
or
2) Determining if a series converges
First, let's look at a special kind of series for which we can find a sum:
This is called a Telescoping Series. Solve by writing out the first
few terms of Sn, then take limit.
Example: 
(use partial fractions, then telescoping)
A Geometric Series is one whose terms are elements of a Geometric Sequence:
Note the sum is
from 0!!
The solution is: 
To derive this, the sum must exist to algebraically manipulate it.....
Examples: 
Summation Rules: const. mult., sum, diff
Rule: if 
converges,
then 
. By
contrapositive,
if 
, then 
diverges.
So this is similar to the second derivative test: if 
, then
we know the series must diverge, but if 
, then we can't
say anything about the convergence of the series.
Examples: 
