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# MacLaurin Series And Laplace Transforms : Absolute Convergence

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Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s)
Determine 5 values for which the series converges absolutley (and uniformly).
Also show the Laplace transform exists, i.e. that it has exponential order alpha.
Here are the functions.

A) f(t)=cosh (bt)
B) f(t)=cos (bt)
C) f(t)=t sin (bt)
D) f(t)=sinh (t)
_______
t

##### Solution Summary

MacLaurin series And Laplace transforms are used to define values for which functions converge absolutely.

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Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s)
Determine 5 values for which the series converges absolutley (and uniformly).
Also show the Laplace transform exists, i.e. that it has exponential order alpha.
Here are the functions. If it is too much work to solve all of them given the number of credit, I have given, then please solve as many as you can and hopefully I can figure out the others.
A) f(t)=cosh (bt)
B) f(t)=cos (bt)
C) f(t)=t sin (bt)
D) f(t)=sinh (t)

Solution.

A) f(t)=cosh (bt)=

We know that , so
and
So,

...............................................................................................(1)

So, the MacLaurin Series for is above (1)

Now ...

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
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• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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