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We want to prove the following statement: Theorem 0.1. Let f (x) be a continuous, nonnegative, decreasing function. Suppose that the integral Z ∞ f (x) dx 1 diverges. Then the sum ∞ X f (n) n=1 also diverges. R∞ Proof. Suppose that 1 f (x) dx diverges. Because f (x) is nonnegative and continuous, this is only possible if the integral diverges to infinity. This means that for every K > 0, we can find a number B such that Z B f (x) dx ≥ K. 1 But the sum B X f (n) n=1 is an upper Riemann sum for this integral (because f is decreasing). So we have that B X f (n) ≥ K. n=1 Because we can do this for any K, it follows that the sum ∞ X f (n) n=1 diverges to ∞. 1