Let Pn-23-123-1-33-133-1-43-143-1-n3-1n3-1- Evaluate limn- Pn-

P_n = 2^3 1/2^3 + 1.3^3 13^3 + 1.4^3 14^3 + 1.n^3 1n^3 + 1 =79.2628.6365.n^3 1n^3 1 =79.2628.6365.n^3( 1 1n^3)n^3( 1 + 1n^3) l ...

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Definite Integral as Limit of Sum

Let Pn=23-1...

Question

Let $P_{n} = \frac{2^{3} - 1}{2^{3} + 1} . \frac{3^{3} - 1}{3^{3} + 1} . \frac{4^{3} - 1}{4^{3} + 1} . \frac{n^{3} - 1}{n^{3} + 1} .$ Evaluate $lim n \to \infty P_{n}$ .

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Solution

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lim n \to \infty \frac{2^{3} - 1^{3}}{2^{3} + 1^{3}} . \frac{3^{3} - 1^{3}}{3^{3} + 1^{3}} . . . . \frac{n^{3} - 1}{n^{3} + 1}

l i m_{n \to \infty} \frac{2^{3} - 1^{3}}{2^{3} + 1^{3}} . \frac{3^{3} - 1^{3}}{3^{3} + 1^{3}} . . . . \frac{n^{3} - 1}{n^{3} + 1}

${lim}_{n \to \infty} \frac{1^{3} + 2^{3} + 3^{3} + . . . . + n^{3}}{(n - 1)^{4}}$

lim n \to \infty \frac{1 + 2^{4} + 3^{4} + . . . + n^{4}}{n^{5}} - lim n \to \infty \frac{1 + 2^{3} + 3^{3} + . . . + n^{3}}{n^{5}}

{lim}_{n \to \infty} \frac{2^{3} - 1^{3}}{2^{3} + 1^{3}} \cdot \frac{3^{3} - 1^{3}}{3^{3} + 1^{3}} \dots \frac{n^{3} - 1}{n^{3} + 1}

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