Question

`sum_(k=1)^oo (k!)/(3k)^k =` ?

Answer 1

See below.

Explanation:

Using the Stirling asymptotic formula

`k! approx sqrt(2pi k)(k/e)^k` we can compute an estimate for the series value

`sum_(k=1)^oo (k!)/(3k)^k approx sqrt(2pi) sum_(k=1)^oo sqrt(k)/(3e)^k` and so

`sum_(k=1)^oo (k!)/(3k)^k le sqrt(2pi) sum_(k=1)^oo k/(3e)^k`

but

`sqrt(2pi) sum_(k=1)^oo k/(3e)^k = sqrt(2pi) (3e)/(3e-1)^2 = 0.399305` then

`sum_(k=1)^oo (k!)/(3k)^k le 0.399305`

The exact value with `10` terms is about `0.398459`

This is a rapidly convergent series so for `k > 10` the significant digits remain.