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 ?
How to show a sequence is monotonically decreasing and a null sequence?
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real-analysis
sequences-and-series
limits
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0What is a null sequence? – 2017-02-14
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0a sequence whose limit to infinty is 0 – 2017-02-14
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1`how do i prove that it is monotonically decreasing` By definition, just prove that $a_{n+1} \le a_n\,$. – 2017-02-14
2 Answers
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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 $0$.
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we compute $$a_{n+1}-a_n=\frac{n+2}{(n+1)^2}-\frac{n+1}{n^2}=\frac{n^2(n+2)-(n+1)^3}{n^2(n+1)^2}=-\frac{n^2+3n+1}{n^2(n+1)^2}<0$$