Example. Expand in a power series at
and find the interval of convergence.
I will use the series for :
I need powers of , so I make an "
" on the bottom, then fix the numbers so the
value of the fraction doesn't change. Then I do algebra to put my
function into the form
, at
which point I can substitute:
I substituted in the
u-series to get my series.
The interval of convergence for the series for is
. Substitute
:
Example. Find the interval of convergence of
.
Apply the Root Test:
The series converges for
At , the series is
This series diverges by the Zero Limit Test.
At , the series is
This series also diverges by the Zero Limit Test.
The power series converges for and diverges elsewhere.
Example. Expand in a power series at
and find the interval of convergence.
Set in
This gives
The interval of convergence for the series is
. So
for the
series,
Example. Expand in a Taylor series at
.
Using the double angle formula
Example. (a) Use the first four nonzero terms
of the Taylor series for at
to approximate
.
(b) Use the Alternating Series Test to estimate the error in part (a).
(a)
Hence,
(I used the first four terms to get the approximation.)
(b) The error is no greater than the next term, which is .
Example. Use the Taylor series expansion of
at
to explain the fact
that
.
The series for at
is
Divide by x to obtain
Then
Example. Find the first four nonzero terms of
the Taylor expansion for at
.
The series is
Example. Find the interval of convergence of
.
The series converges for
At , the series is
This is harmonic, so it diverges.
At , the series is
This is harmonic, so it diverges.
The power series converges for and diverges
elsewhere.
Example. satisfies
Use the third degree Taylor polynomial for f at to approximate
.
The third degree Taylor polynomial for f at is
So
Example. Suppose that . Use
to estimate the error in using the fourth degree
Taylor polynomial at
to approximate
for
.
For some z between 0 and x,
Since ,
.
For the z-term, I have . Thus,
So .
Therefore,
Copyright 2019 by Bruce Ikenaga