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匿名使用者 發問時間: 科學數學 · 1 0 年前

高等微積分*=_=*

Prove that the following sequences are uniformly convergent for the range of x given:(c)log(1+nx)/n, 1<=x<=2(d)n/e^nx^2, 1/2<=x<=1(過程請詳答)

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  • 1 0 年前
    最佳解答

    Qc.

    f_n(x)= log[(1+nx)/n] -> log(x)

    For all ε>0, take N> 1/(10^ε - 1) (indep. on x).

    If n>N, then

    | f_n(x)- log(x)| = | log[(1+nx)/(nx)] | = | log[1+ 1/(nx)] |

    <= log(1+1/n)

    < log(1+ 1/N)= log(1+ 10^ε - 1)= ε

    故 f_n(x)= log[(1+nx)/n] -> log(x) uniformly.

    註:若log(x)為自然對數, 則以上證明為10,請改為e

    Qd.

    因e^x = 1+x+x^2/2!+ x^3/3!+...> x^2 /2 (for x>0)

    => n/e^(nx^2) < n/( n^2 x^4/ 2) < 32/n

    設f_n(x)= n/e^(nx^2), 1/2 <= x <= 1

    For all ε>0, take N> 32/ε (indep. on x).

    If n>N, then | f_n(x) - 0 | < 32/n < 32/N< ε.

    故 f_n(x) -> 0 uniformly.

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