`sum_(n=1)^oo (n!)/2^n` Verify that the infinite series diverges

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`sum_(n=1)^oo (n!)/2^n`

To verify if the series diverges, apply the ratio test. The formula for the ratio test is:

`L = lim_(n->oo) |a_(n+1)/a_n|`

If L<1, the series converges.

If L>1, the series diverges.

And if L=1, the test is inconclusive.

Applying the formula above, the value of L will be:

`L = lim_(n->oo) |(((n+1)!)/2^(n+1))/ ((n!)/2^n)|`

`L= lim_(n->oo) |((n+1)!)/2^(n+1) * 2^n/(n!)|`

`L=lim_(n->oo) | ((n+1)*n!)/(2*2^n) * 2^n/(n!)|`

`L = lim_(n->oo) | (n+1)/2|`

`L = 1/2 lim_(n->oo) |n+ 1|`

`L=1/2 * oo`

`L=oo`

Therefore, the series diverges.

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