ISI-MMA 2019 Sample Questions-23

Question

For $n \geq1$, Let

$a_{n} = \frac{1}{2^{2}} + \frac{2}{3^{2}} +.....+ \frac{n}{(n+1)^{2}}$ and $b_{n} = c_{0} + c_{1}r + c_{2}r^{2}+.....+c_{n}r^{n},$

where$|c_{k}| \leq M$ for all integers $k$ and $|r| \leq 1.$ Then

(A) both $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences

(B) $\{a_n\}$ is a Cauchy sequence but $\{b_n\}$ is not a Cauchy sequence

(C) $\{a_n\}$ is not a Cauchy sequence but $\{b_n\}$ is a Cauchy sequence

(D) neither $\{a_n\}$ nor $\{b_n\}$ is a Cauchy sequence.

Comments

@Kushagra Chatterjee could you please help ?

Answer 1

Option C

Note:Any sequence is cauchy sequence iff the sequence is convergent. 

Comments

Thanks :)