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Explanation:

We need to check options.

Option (a): $\frac{\mathrm{\pi }}{2}\left[\frac{1}{\mathrm{\pi }}+1\right]-\frac{\mathrm{\pi }}{2}$
= $\frac{1}{2}+\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{2}$ = $\frac{1}{2}$ = 0.5 (terminating)

Option (b): sin21° + sin22° + … + sin288° + sin289°
= sin21° + sin22° + …  +  sin245° + … + cos22° + cos21°
= 44 + ½ = 44.5 (terminating)

Option (c): $\frac{\sqrt[3]{729}}{3}+\frac{22}{7}$ 
= $\frac{9}{3}+\frac{22}{7}$ = 3 + 22/7 (terminating)

Option (d): $\left(4-\mathrm{\pi }\right)[1+\frac{\mathrm{\pi }}{4}+{\left(\frac{\mathrm{\pi }}{4}\right)}^2+{\left(\frac{\mathrm{\pi }}{4}\right)}^3+... inifinite terms]$
= $\left(4-\mathrm{\pi }\right)\left[\frac{1}{1-{\displaystyle \frac{\mathrm{\pi }}{4}}}\right]$ = 4 (non-terminating)

Option (e): $\sqrt{2}\left(3\sqrt{2}-\frac{4}{\sqrt{2}}\right)+\sqrt{3}$
= 6 - 4 + √3 = 2 + √3 (non-terminating and non-repeating)

Hence, option (e).