It was a question in a calculus textbook and I didn't know how he did the algebra between these steps

$$ \lim_{n\to \infty} \frac{81}{4}\left(\frac{n(n+1)}{2}\right)^2-\frac{54}{n^2}\frac{n(n+1)}{2} $$

$$ \lim_{n\to \infty} \frac{81}{4}\left(\frac{1}{n}+1\right)^2-27\left(\frac{1}{n}+1\right) $$

How did he go from the first to the second step?

  • $\begingroup$ You mean $n\to\infty$, right? $\endgroup$ – draks ... Nov 22 '13 at 11:39
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    $\begingroup$ Limit should be on $n$ first part is wrong. $\endgroup$ – lab bhattacharjee Nov 22 '13 at 11:39
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    $\begingroup$ $\dfrac{54}{2}=27$; $\dfrac{n(n+1)}{n^2}=\dfrac{1}{n}+1$. But, as written, the two expressions are not equal. $\endgroup$ – egreg Nov 22 '13 at 11:42

HINT: $\hskip1.7in\displaystyle\frac{n(1+n)}{n^2}=\frac{n+n^2}{n^2}=\frac n{n^2} + \frac{n^2}{n^2}=\frac1n+1,$

so it looks like there's a $n^4$ ($n^2$ inside the square's bracket) missing in the first denominator...


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