Re: [微積] 請教幾題微積分
※ 引述《light0617 (期待寒假)》之銘言：
: 請教幾題微積分
: n
: 1.LIM [ 1/(n^0.5)]*sigma(1/(k^0.5))
: k=1
: n->無限大
1 n 1
lim (----)(Σ -----)
n→∞ √n k=1 √k
1 n 1
= lim (---)(Σ ---------)
n→∞ n k=1 √(k/n)
1 1
= ∫ ----- dx
0 √x
|1
= (2)(√x)| = 2
|0
: 4x
: 2.lim x*S (t^6+100)^(-1/3)dt
: x->無限大 2x
4x
lim (x)(∫ (t^6 + 100)^(-1/3) dt)
x→∞ 2x
4x
∫ (t^6 + 100)^(-1/3) dt
2x
= lim ------------------------------
x→∞ 1/x
(4)(((4x)^6 + 100)^(-1/3)) - (2)(((2x)^6 + 100)^(-1/3))
= lim ---------------------------------------------------------
x→∞ -1/(x^2)
-x^2
= lim -----------------------------------------------------
x→∞ (4)(((4x)^6 + 100)^(1/3)) - (2)(((2x)^6 + 100)^(1/3))
-1
= lim ----------------------------------------------------------
x→∞ (4)((4^6 + 100/(x^6))^(1/3) - (2)((2^6 + 100/(x^6))^(1/3))
-1 -1
= ------------- = ----
4*4^2 - 2*2^2 56
: -1
: 3.F(x)=S (x^3-t)f(t^(1/3))dt X<0
: x^3
: where f(x) is a continous function.Suppose F(x) is an antiderivative of x^4
: ,find F(x) and f(x)
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