At what point do the sums of the reciprocals converge?

 P: 2,179 The integral test for convergence tells us that the infinite sum: $$\sum_1^\infty f(n)$$ and the integral: $$\int_1^\infty f(n)$$ either both converge or both diverge. Since: $$\int_1^\infty \frac{1}{x^{1+\delta}} = \frac{1}{\delta}$$ This tells us that the infinite series: $$\sum_1^\infty \frac{1}{x^{1+\delta}}$$ will converge as long as delta is greater than zero, no matter how small it is.