Question

it is possible to have both Summation a_n and Summation b_n be divergent series and yet have Summation from n=1 to inf (a_n +- b_n) be a convergent series.
What does this statement means? Does it mean that if both Summation a_n and Summation b_n are divergent series then is it possible to Summation from n=1 to inf (a_n +- b_n) to be a convergence?

Answer 1

if \(a_n=1\) and \(b_n=-1\)
\[\sum_{n=1}^{\infty}a_n\text{ diverges}\]
and
\[\sum_{n=1}^{\infty}b_n\text{ diverges}\]

but

\[\sum_{n=1}^{\infty}(a_n+b_n)=0\]

Answer 2

OK