ANSWER :∑n=1∞an=∑n=1∞n(−1)n,bn=(−1)n
EXPLANATION The series ∑n=1∞n(−1)n is an alternating series, since the sequence an=(−1)ncn and cn=n1 is decreasing , limn→∞cn=0 . The series ∑n=1∞n1 diverges, because it is a p− series for p=1 , hence the series ∑n=1∞n(−1)n converges conditionally . The sequence bn=(−1)n is a bounded sequence , since −1≤bn≤1 for all n∈N (b2n=1,b2n−1=−1 ). Since an⋅bn=n1 , then the series ∑n=1∞an⋅bn=∑n=1∞n1 diverges.
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