Computing summations and integrals using indefinite summation

Given a summation of a function from, say, zero to N, one may consider analytically continuing the upper limit N to the set of reals or the complex plane. Consider such an analytic continuation for the summation 1 + 2 + 3 +...+N, without invoking the well known result of this summation. Argue that the summation is a second degree polynomial, find its zeros and determine the result this way. Repeat this for the summation of the squares 1^2 + 2^2 + 3^2 +...+N^2, by arguing on the basis that this should be a third degree polynomial.

Evaluate the integral from 0 to infinity of exp(-x^2)/(1+x^2)dx using Glaisher's theorem.

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