Assignment:
By considering appropriate series expansions, prove that the power series expansion of the product of the (infinitely many) exponential factors e^{(x^i)/i}, i = 1, 2, 3, ..., is 1 + x + x^2 ... for |x| <1.
By expanding each individual exponential factor in the product and multiplying out, also show that the coefficient of x^19 in the power series expansion of the product has the form
1/(19!) + 1/19 + r/s
where 19 does not divide s. Deduce that
18! = -1 (mod 19).
Provide complete and step by step solution for the question and show calculations and use formulas.