# MacLaurin Series And Laplace Transforms : Absolute Convergence

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

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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 ...

#### Solution Summary

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