Name:
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Math 122 Lab 4: Graphical
Power Series
Introduction
You
have introduced yourself to power series. At this point, it may still be a
little mysterious. One goal of this lab is to construct visual images of
several power series to help you understand them.
What to Turn In
Turn
in this worksheet with everything filled in. Be sure to sketch all graphs, and
label them! This will be graded for accuracy and understanding, so write good
explanations when you are asked to explain something. Due
Friday. Late papers lose 4 points per day.
A Geometric
Power Series
This is the easiest type of series for us to handle.
To be specific, we will work with
which has
an interval of convergence of (-1,1). Start by graphing 
and 
on the same graph with
graphing window -2<x<2 and -4<y<10. For what range of x-values are
the graphs very close together?
Next, graph and 
on the same graph. For what range of x-values are the graphs very close
together? What is different about the way the cubic approximates
?
Next, graph and 
on the same graph. For what range of x-values are the graphs very close
together?
Finally, graph and 
on the same graph. Briefly discuss what is meant by “the interval of
convergence is (-1,1)”. Looking at the function , why would it be impossible for the radius of convergence to
be larger than 1?
A Power Series with Center Moved to
x=1
Next, we will work with
Start
by graphing 
and 
on the same graph with
graphing window -1<x<3 and -4<y<10. For what range of x-values are
the graphs very close together?
Similar
to your previous work, sketch two more graphs with more terms added and use the
graphs to conjecture the radius of convergence.
A More Complex Power Series
Write
out a power series (center 0) for the function 
.
Similar
to what you have done before, sketch graphs showing the original function and
approximations with (a) 3 terms and (b) 6 terms. Use an appropriate graphing
window. Based on the graphs, conjecture the radius of convergence.
To
see why there is a barrier to convergence, solve the equation 
for complex solutions
(this gives complex “vertical asymptotes”). A complex number a+bi is graphed as the point (a,b). Thinking of your solution(s) graphically, what
is the distance between your solution and the origin? (Note: the full theory of
power series requires complex variables to make complete sense.)
You
will see in section 7.7 that 
Sketch several graphs and use them to conjecture the radius
of convergence.
You
will see in section 7.7 that 
Sketch several graphs
and use them to conjecture the radius of convergence.
According
to your graphs, is the approximation 
very accurate at x = 1.1? Substitute x = 1.1 into the right-hand side and compare the value to the
calculator’s value of ln(1.1). Does power series actually give us an easy way to
compute difficult functions?
.