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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