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

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Report 5

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Problem 1: Taylor Series Expansion of the log Function

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Problem Statement

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Use the point

Solution

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Set













For the series expansion results in,


Plots of taylor series expansion: Up to order 4


Up to order 7


Up to order 11


Up to order 16


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

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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

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Number 1

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Plot at least 3 truncated series to show convergence


m=0:

m=1:

m=2:



Number 2

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Plot at least 3 truncated series to show convergence


m=0:

m=1:

m=2:


Number 3

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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

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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

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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

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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

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


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

Number 8

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




Cancelling some terms out, you get

Using L'Hopitals Rule, you get

Number 9

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





Cancelling some terms out, you get

Using L'Hopitals Rule, you get

Number 10

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




Cancelling some terms out, you get

Using L'Hopitals Rule, you get

Number 11

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



Honor Pledge

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

Problem 3:

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Problem Statement

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



Solution

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Determinant of the Matrix of Components
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The Matrix of components of the vectors and is



So the vectors and are linearly independent.

Gramian
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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 | edit source]

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

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Problem Statement

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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

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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 | edit source]

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