Answer to Question #169568 in Real Analysis for Akshay

Question

Check whether the following sequences (sn) are Cauchy, where (i)sn=1+2+3+...+n 4n3 + 3n (ii) sn 3n + n2 

Answer 1

Solution:

We know that every cauchy sequence is convergent. So we check convergence for given sequences.

(i) $S_n=1+2+3+...+n=\dfrac{n(n+1)}2$

By series convergent test, we know that this series diverges.

Hence, this one is not cauchy sequence.

(ii) $S_n=\dfrac{4n^3+3n}{3n+n^2}$

$S_n=\dfrac{n^3(4+\frac{3}{n^2})}{n^2(\frac{3}{n}+1)}$

$S_n=\dfrac{n(4+\frac{3}{n^2})}{(\frac{3}{n}+1)}$

$\lim_{n\rightarrow \infty}S_n=\dfrac{\infty(4+0)}{(0+1)}=$ diverges

Hence, this one also is not cauchy sequence.