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# Answer to Question #2334 in Real Analysis for Joanna

Question #2334
If &sum; an with an &gt; 0 is convergent, then is &sum; (an an+1)1/2 always convergent? Either prove it
or give a counterexample.

If we consider two convergent series , for which &sum;(a(n))= A, and &sum;(b(n)) = B, then the following also converge as indicated:
&sum;(a(n)+b(n)) = A + B
&sum;( (a(n)+a(n+1))/2) =& 0.5(&sum;(a(n) + &sum;(a(n+1))) = 0.5&sum;(a(n))+0.5 &sum; (a(n+1)) .
Whereas the &sum;(a(n)) converges absolutely (a(n)&gt;0) , then &sum;( (a(n)+a(n+1)/2)& converges too because at in right part of the equation there are two convergent series.

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