Testing for Sequence Convergence with Surds

Multiply numerator and denominator by $\sqrt {n^{2}+5n}+\sqrt {n^{2}+2n}$ to see that $a_n=\frac {3n} {\sqrt {n^{2}+5n}+\sqrt {n^{2}+2n}}$. Divide all the terms by $n$ to see that the limit is $\frac 3 2$.


Using the binomial series, $$n\left(1+\dfrac5n\right)^{1/2}-n\left(1+\dfrac2n\right)^{1/2}=n\left(1+\dfrac12\dfrac5n+O\left[\dfrac1{n^2}\right]\right)-n\left(1+\dfrac12\dfrac2n+O\left[\dfrac1{n^2}\right]\right)\to\dfrac32.$$

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Limits

Sequences And Series

Convergence Divergence

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