Review of
Power Series
and Taylor Polynomials
Infinite Sums and
Power Series
Recall Infinite
Sums:
Infinite Sums and
Power Series
Recall Infinite
Sums:
Infinite Sums and
Power Series
Recall Infinite
Sums:
In General:
Infinite Sums and
Power Series
In General:
Three possible outcomes of infinite sums:
• Diverges
• Converges
• Neither
or
Infinite Sums and
Power Series
In General:
Special Type of Infinite Sum: Power
Series
Which gives
Infinite Sums and
Power Series
In General:
Special Type of Infinite Sum: Power
Series
Depends on x
Infinite Sums and
Power Series
In General:
Special Type of Infinite Sum: Power
Series
Depends on x
Infinite Sums and
Power Series
In General:
Special Type of Infinite Sum: Power
Series
Depending on x, can either diverge, converge, or neither
Example
First 10
Terms
Example
First 20
Terms
Example
First 50
Terms
Example
First 100
Terms
Example
Eventuall
y
Determining Interval of
Convergence
Ratio
Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of
Convergence
Now Note If:
The
n
Determining Interval of
Convergence
Ratio
Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of
Convergence
Ratio
Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of
Convergence
Ratio
Test
Converges if
Diverges if
If
, test is inconclusive.
Determining Interval of
Convergence
Ratio
Test
Converges if
Diverges if
If
, test is inconclusive.
Is Known as
“Radius of
Convergence”
Determining Interval of
Convergence
Radius of Convergence
Interval of Convergence
Power Series Manipulation
Sum
:
Derivative:
Reindexing:
Taylor Series
For a Power
Series
It is always true that
So given a function, we can write it as a power series.
A power series that describes a function is called a
“Taylor Series”
Taylor Series
For a Power
Series
For
Example
Taylor Series
Taylor Series
Taylor Series
Taylor Series
Taylor Polynomials
If we kept going, we would find
no error!
KEY IDEA: We can use
Taylor Series (or Power Series)
as substitutes for common functions, because
they ARE THE SAME THING
Questions?