Crazy Limits!!! 1

Calculus Level 5

Let

I=limx0(1x50xet2dt1x4+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)

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) .

Then find

π2×I+9K24.\displaystyle \pi^{2} \times I + \frac{9K^{2}}{4}.

Details and Assumptions

  • [x][x] denotes the floor function.

  • Use π2=10\pi^2 = 10.

×

Problem Loading...

Note Loading...

Set Loading...