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Let
I=limx→0(1x5∫0xe−t2dt−1x4+13x2)\displaystyle I = \lim_{x \to 0}\left(\frac{1}{x^{5}} \int_0^{x} e^{-t^{2}}dt -\frac{1}{x^{4}} + \frac{1}{3x^{2}} \right) I=x→0lim(x51∫0xe−t2dt−x41+3x21)
and
K=limn→∞([127]+[227]+…+[n27]n3).\displaystyle K= \lim_{n \to \infty}\left(\frac{[1^{2}\sqrt{7}] + [2^{2}\sqrt{7}]+ \ldots + [n^{2}\sqrt{7}]}{n^{3}} \right) .K=n→∞lim(n3[127]+[227]+…+[n27]).
Then find
π2×I+9K24.\displaystyle \pi^{2} \times I + \frac{9K^{2}}{4}.π2×I+49K2.
Details and Assumptions
[x][x][x] denotes the floor function.
Use π2=10\pi^2 = 10π2=10.
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