Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :

limk→∞ [13 + 23 + ……. + k3] / [k4] =

1) 0

2) 2

3) 1/3

4) ∞

5) 1/4

Solution: (5) ¼

\(\begin{array}{l}\begin{array}{l} \lim _{k \rightarrow \infty}\left(\frac{1^{3}+2^{3}+\cdots+k^{3}}{k^{4}}\right) \\ \Rightarrow \lim _{k \rightarrow \infty} \frac{\left[\frac{k(k+1)}{2}\right]^{2}}{k^{4}} \\ \Rightarrow \lim _{k \rightarrow \infty} \frac{k^{2}(k+1)^{2}}{4 k^{4}} \\ \Rightarrow \frac{1}{4} \lim _{k \rightarrow \infty} \frac{(k+1)^{2}}{k^{2}} \\ \Rightarrow \frac{1}{4} \lim _{k \rightarrow \infty}\left(\frac{k+1}{k}\right)^{2} \\ \text { As } \mathrm{k} \rightarrow \infty, \frac{1}{k} \rightarrow 0 \\ \Rightarrow \frac{1}{4} \lim _{k \rightarrow \infty}\left(1+\frac{1}{k}\right)^{2} \\ \Rightarrow \frac{1}{4}(1)^{2}\\=\frac{1}{4} \end{array}\end{array} \)

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