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Another proof that the harmonic series diverges
If the harmonic series converges, then its kth ‘tail’
∞
X
1
T (k) :=
n
n=k+1
is well-defined, and is a strictly decreasing function of k. Now note that
T (2m) =
1
1
1
1
+
+
+
+ ···
2m + 1 2m + 2 2m + 3 2m + 4
≥
1
1
1
1
+
+
+
+ ···
2m + 2 2m + 2 2m + 4 2m + 4
=
2
2
+
+ ···
2m + 2 2m + 4
=
1
1
+
+ ···
m+1 m+2
= T (m),
and if m ≥ 1 then this contradicts that T is a strictly decreasing function.
1
new.pdf (PDF, 79.12 KB)
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