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Question Number 476 by 123456 last updated on 11/Jan/15
given a_n  and b_n  two real sequence  can a serie Σ_(n=1) ^(+∞) a_n  and Σ_(n=1) ^(+∞) b_n  diverge  but  Σ_(n=1) ^(+∞) (a_n +b_n ) converge?
$${given}\:{a}_{{n}} \:{and}\:{b}_{{n}} \:{two}\:{real}\:{sequence} \\ $$$${can}\:{a}\:{serie}\:\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}{a}_{{n}} \:{and}\:\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}{b}_{{n}} \:{diverge} \\ $$$${but} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\left({a}_{{n}} +{b}_{{n}} \right)\:{converge}? \\ $$
Commented by prakash jain last updated on 11/Jan/15
a_n =2^n   b_n =−2^n
$${a}_{{n}} =\mathrm{2}^{{n}} \\ $$$${b}_{{n}} =−\mathrm{2}^{{n}} \\ $$

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