Definite integral with summation

Calculus Level pending

If

\[ \ { I }_{ n }=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { x }^{ n }\cos { x } dx } \]

Then

\[ \sum _{ n=2 }^{ \infty }{ \left( \frac { { I }_{ n } }{ n! } +\frac { { I }_{ n-2 } }{ \left( n-2 \right) ! } \right) } \]

is of the form \( { e }^{ a }-b-1 \) , find the value of \(a-b\).

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