1
$\begingroup$

For example Let the sequence be $a_n=\frac{n+1}{n^2}$. I proved that $a_n$ is a null sequence by factoring out the $n^2$ .My question is how do i prove that it is monotonically decreasing? . Do i find the limit of the ratio of $\frac{a_{n+1}}{a_n}$ to infinity. Or do i show that $a_n$ is Cauchy ?

$\endgroup$
  • $\begingroup$ What is a null sequence? $\endgroup$ – Umberto P. Feb 14 '17 at 19:58
  • $\begingroup$ a sequence whose limit to infinty is 0 $\endgroup$ – asddf Feb 14 '17 at 19:58
  • 1
    $\begingroup$ how do i prove that it is monotonically decreasing By definition, just prove that $a_{n+1} \le a_n\,$. $\endgroup$ – dxiv Feb 14 '17 at 19:58
3
$\begingroup$

By writing $$ a_n=\frac1n+\frac1{n^2}, \qquad n\ge1, $$ one sees that $\{a_n\}$ is monotonically decreasing to $0$ being the sum of two monotonically decreasing sequences to $0$.

$\endgroup$
1
$\begingroup$

we compute $$a_{n+1}-a_n=\frac{n+2}{(n+1)^2}-\frac{n+1}{n^2}=\frac{n^2(n+2)-(n+1)^3}{n^2(n+1)^2}=-\frac{n^2+3n+1}{n^2(n+1)^2}<0$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.