lim_n→∞( ∑_r=1^n12n + (2n 1)) = lim_n→∞( ∑_r=1^n14n 1) = lim_n→∞( 1n∑_r=1^nn4n 1) = lim_n→∞( 1n.n4n 1∑_r=1^n (1)) = lim_n→∞( 1n.n ...

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Rationalization Method to Remove Indeterminate Form

limn →∞∑r=1n1...

Question

$lim n \to \infty (n \sum r = 1 \frac{1}{2 n + (2 n - 1)})$

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Solution

$lim n \to \infty (n \sum r = 1 \frac{1}{2 n + (2 n - 1)})$

$= lim n \to \infty (n \sum r = 1 \frac{1}{4 n - 1})$

$= lim n \to \infty (\frac{1}{n} n \sum r = 1 \frac{n}{4 n - 1})$

$= lim n \to \infty (\frac{1}{n} . \frac{n}{4 n - 1} n \sum r = 1 (1))$

$= lim n \to \infty (\frac{1}{n} . \frac{n}{4 n - 1} . n)$

$= lim n \to \infty (\frac{n}{4 n - 1})$

$= lim n \to \infty ⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{4 - \frac{1}{n}} ⎞ ⎟ ⎟ ⎟ ⎠$

$= \frac{1}{4 - 0}$

$= \frac{1}{4}$ (Ans)

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