Okay, I have been trying and I am not sure if I am on the right track, somebody please kindly enlighten me. Thank you.

This is what I got so far, I converted sinh 2x back to its basic definition,

Then I differentiated the basic definition instead.

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Further simplifying,

Therefore, the Taylor series associated with the function f(x) is

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For part (b), to prove with ratio test that this series is convergent means

How should I proceed from here?

For part (c), to show that the Taylor series represents f(x) exactly in the interval in part (b) above means that I'll have to prove that the remainder is equivalent to zero as right?

I don't know how to proceed from here either.

Thank you for your time!!