THE SUM OF TWO DIVERGENT SERIES IS NOT NECESSARILY A DIVERGENT
SERIES
Conjecture: If both
+∞
X
an
n=n0
and
+∞
X
bn
n=n0
are divergent, then
+∞
X
(an + bn )
n=n0
is also divergent.
The conjecture is false. We will prove this by the following counterexample:
Let an = √1n , cn = n12 , and bn = cn − an .
+∞
X
an
n=1
is divergent, because it is a p-series with p =
1
2
≤ 1.
+∞
X
cn
n=1
is convergent, because it is a p-series with p = 2 > 1.
Then
+∞
+∞
+∞
+∞
X
X
X
X
an
cn −
(cn − an ) =
bn =
n=1
n=1
n=1
n=1
is divergent as the difference of a convergent and a divergent series. So, even if both
+∞
X
an
n=1
and
+∞
X
bn
n=1
are divergent,
+∞
X
n=1
(an + bn ) =
+∞
X
(an + cn − an ) =
n=1
+∞
X
n=1
is convergent, not divergent.
This concludes the proof that the given conjecture is false.
1
cn