riemann sum

👨🏻‍🏫 學術台數學

子路

2009-12-20 4:15 am

as follows~~~~~~~~~~~~~~

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更新1:

STEVIE-G™ you get the point it sums from j=1 to n^2, not n

回答 (3)

myisland8132

2009-12-20 7:16 am

✔ 最佳答案

lim Σ n/(n^2+j^2)
=lim Σ (1/n)[n^2/(n^2+j^2)]
=lim Σ (1/n)[1/(1+(j/n)^2)]
=∫ 1/(1+x^2) dx
=tan^(-1) x
=pi/2

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用戶9112

2009-12-21 7:06 am

lim Σ (1/n)[1/(1+(j/n)^2)]
when j = 1, the fraction is 1 / (1 + x^2) and x = (1/n)
when j = n^2, the fraction is 1 / (1 + x^2) and x = (n^2/n) = n
So the upper limit is n (infinity) the lower limit is 1/n (0)
atan(infinity) - atan(0) = pi/2

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STEVIE-G™

2009-12-20 8:21 pm

What's the upper/lower limit of the definite integral?

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