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Real Analysis Question on Sequences and Series

Let an=sin⁡(1n3)a_n=\sin(\frac{1}{n^3}) and bn=sin⁡(1n)b_n=\sin(\frac{1}{n}) for n ∈ N\N. Then

A

both ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n and ∑n=1∞bn\sum\limits_{n=1}^{\infin}b_n are convergent

B

∑n=1∞an\sum\limits_{n=1}^{\infin}a_n is convergent ∑n=1∞bn\sum\limits_{n=1}^{\infin}b_n is NOT convergent

C

∑n=1∞an\sum\limits_{n=1}^{\infin}a_n is NOT convergent ∑n=1∞bn\sum\limits_{n=1}^{\infin}b_n is convergent

D

both ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n and ∑n=1∞bn\sum\limits_{n=1}^{\infin}b_n are NOT convergent

Answer

∑n=1∞an\sum\limits_{n=1}^{\infin}a_n is convergent ∑n=1∞bn\sum\limits_{n=1}^{\infin}b_n is NOT convergent

Explanation

Solution

The correct option is (B) : ∑n=1∞an\sum\limits_{n=1}^{\infin}a_n is convergent ∑n=1∞bn\sum\limits_{n=1}^{\infin}b_n is NOT convergent.