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Lecture 26

# MATH 2B Lecture Notes - Lecture 26: Ratio Test, Bmw 1 Series, Direct Comparison TestPremium

Department
Mathematics
Course Code
MATH 2B
Professor
ERJAEE, G.
Lecture
26

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MATH 2B - Lecture 26 - Power Series
A power series is a series of the form
 
Here x is the variable, is the coefficient
The main thing we are going to do is to find the values of x for
which the power series is convergent
Example:
We take for all n, the power series becomes
 
We know that

  Geometric series

Therefore, the power series
 is convergent when |x|<1
More general, we can have
  which is called a power
series centered at a
Example:
, for what values of x, does this power series converge
Solution:
Again, this is a geometric series

series is convergent
For most of the power series, we use the ratio test or root test to find for which
values the series is convergent
Example:
For what values of x does the series
converge?
Solution:
Ratio test

{ , 
Therefore, the power series
is convergent when x = 0