Mathematica 教學： Prove that 1 - 3 s^2 + (1 + s^2) Cosh[s] >0.

Plot[1-3s^2+(1+s^2)Cosh[s],{s,0,Pi},Frame->True]
SeriesCoefficient[1-3s^2+(1+s^2)Cosh[s],{s,0,n}]
Normal[Series[1-3s^2+(1+s^2)Cosh[s],{s,0,5}]]
Minimize[2-(3s^2)/2+(13s^4)/24,s]
Solve[D[2-(3s^2)/2+(13s^4)/24,s]==0,s]

Since the coefficient in the n-th order of the taylor series of 1-3s^2+(1 + s^2)Cosh[s] is positive for every even term and is
zero for every odd term for all n>=3, it is obvious to see that 1-3s^2+(1+s^2)Cosh[s]>2-(3s^2)/2+(13s^4)/24>25/26, which implies that 1-3s^2+(1+s^2)Cosh[s]>0. This completes the proof.

Tags: Mathematica 教學 微積分, Teaching

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