limits

👨🏻‍🏫 學術台數學

[匿名]

2011-02-20 6:12 am

lim x->1 (x+x^2+x^3+...x^n -n) / (x-1)

更新1:

in terms of n

更新2:

how (x+x^2+x^3+...x^n -n) / (x-1) go to (1+2x+3x^2+...nx^(n-1) ) ??

回答 (3)

螞蟻雄兵

2011-02-20 6:26 am

✔ 最佳答案

lim( x->1)_(x+x^2+x^3+...x^n -n) / (x-1) 0/0 type=lim( x->1)_(1+2x+3x^2+...+nx^(n-1))/1=1+2+3+,,,+n=n(n+1)/2

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用戶10012

2011-02-20 7:48 pm

You don't know what they are doing if you don't know L'Hopital's Rule.
lim g(x)/f(x) = lim g'(x)/f'(x).

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myisland8132

2011-02-20 6:29 am

Treat n as a constant

lim x->1 (x+x^2+x^3+...x^n -n) / (x-1)

= lim x->1 (1+2x+3x^2+...nx^(n-1) )

= 1 + 2 + 3 + ... + n

= n(n+1)/2

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