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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


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