Find the value of the power series $~~\displaystyle\underset{n=0}{\overset{\infty }{\mathop \sum }}\,-\frac{{3{\left( -1 \right)}^{n}}{{x}^{4n+2}}}{4\left( 2n+1 \right)!}~~$ when $~x=\displaystyle\sqrt{\frac{\pi}{2}}\,$.
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