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.
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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 ∞.
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