Re: [微積] 一題級數
※ 引述《cheesesteak (Terry)》之銘言：
: n x(1-x)
: (a)Evaluate the limit f(x)=lim Σ —————, where 0≦x≦1.
: n→∞ k=1 k+(n-k)x
: (b)Find the extreme values of f(x) on [0,1].
n 1
Note that f(x) = x Σ ------------
k=1 (n/1-x) + k
Let c= n/(1-x) and notice that:
n 1 n dy n
(1) Σ ----- < S ----- = ln (1+ -) = ln(2+x)
k=1 c+k 0 c+y c
n 1 n dy n n(1-x)
(2) Σ ----- > S ----- = ln (1+ ---) = ln(1+ -------) -> ln(2+x)
k=1 c+k 0 c-1+y c-1 n-1+x
So:
f(x) -> xln(2+x)
The extreme value could then be achieved via differentiating.
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