University of Florida/Egm4313/IEA-f13-team10/R5

From Wikiversity
Jump to navigation Jump to search

Report 5[edit]

Problem 1: Taylor Series Expansion of the log Function[edit]

Problem Statement[edit]



Use the point

Solution[edit]





Set













For the series expansion results in,


Plots of taylor series expansion: Up to order 4
Taylor1.JPG

Up to order 7
Taylor2.JPG

Up to order 11
Taylor3.JPG

Up to order 16
Taylor4.JPG

The visually estimated domain of convergence is from .8 to .2.
Now use the transformation of variable


If has a domain of convergence from then converges from

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.


Problem 2: Plots of Truncated Series[edit]

Number 1[edit]

Plot at least 3 truncated series to show convergence


m=0:

m=1:

m=2:


P2G.png


Number 2[edit]

Plot at least 3 truncated series to show convergence


m=0:

m=1:

m=2:


P1G2.png

Number 3[edit]

Find the radius of convergence for the taylor series of sinx, x = 0

The Taylor series of sinx is:



The radius of convergence can be found by:

Number 4[edit]

Find the radius of convergence for the taylor series of log(1+x), x = 0

The Taylor series of log(x+1) is:

The radius of convergence can be found by:



Number 5[edit]

Find the radius of convergence for the taylor series of log(1+x), x = 1

The Taylor series of log(x+1) is:

The radius of convergence can be found by:


Number 6[edit]

derive the expression for the radius of convergence of log(1+x) about any focus point

The taylor series of log(1+x) is:


Number 7[edit]

Find the Taylor series representation of log(3+4x)


Expanding out 4 terms results in,
[
The series representation is

Number 8[edit]

Radius of convergence of log(3+4x) about the point




Cancelling some terms out, you get

Using L'Hopitals Rule, you get

Number 9[edit]

Radius of convergence of log(3+4x) about the point





Cancelling some terms out, you get

Using L'Hopitals Rule, you get

Number 10[edit]

Radius of convergence of log(3+4x) about the point




Cancelling some terms out, you get

Using L'Hopitals Rule, you get

Number 11[edit]

Radius of convergence of log(3+4x) about any given point



Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 3:[edit]

Problem Statement[edit]

Use the Determinant of the Matrix of Components and the Gramian to verify the linear independence of the two vectors and .



Solution[edit]

Determinant of the Matrix of Components[edit]

The Matrix of components of the vectors and is



So the vectors and are linearly independent.

Gramian[edit]

For vectors, the Gramian is defined as:

where:



For the given vectors, the dot products are:






So the Gramian matrix becomes:

Finding the determinant of the Gramian matrix gives the Gramian:

So the vectors and are linearly independent.

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 4: Wronskian and Gramian[edit]

Problem Statement[edit]

Use both the Wronskian and the Gramain to find whether the following functions are linearly independent. Consider the domain of these functions to be [-1, +1] for the construction of the Gramian matrix.





Solution[edit]

Wronskian:



Function is linearly independent if

1)



so function is linearly independent.

2)



so function is linearly independent.


Gramian:


Function is linearly independent if

1)




so function is linearly independent.



2)




so function is linearly independent.


Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.