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Real Analysis Question on Sequences and Series

Let an=1+2−2+...+n−2na_n=\frac{1+2^{-2}+...+n^{-2}}{n} for n ∈ N\N. Then

A

both the sequence (an) and the series ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n are convergent

B

the sequence (an) is convergent but the series ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n are NOT convergent

C

both the sequence (an) and the series ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n are NOT convergent

D

the sequence (an) is NOT convergent but the series ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n is convergent

Answer

the sequence (an) is convergent but the series ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n are NOT convergent

Explanation

Solution

The correct option is (B) : the sequence (an) is convergent but the series ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n are NOT convergent.