Question

高数，第三小题是怎么得到答案的，我这么写对吗？

第八大题的第三小题

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Answer 1

f(x) =lnx => f(1) =0
f'(x) =1/x =>f'(1)/1! =1
f''(x) =-1/x^2 =>f''(1)/2! =-1/2
...
f^(n)(x) = (-1)^(n-1) .(n-1)!/x^n =>f^(n)(1)/n! =(-1)^(n-1)/n
lnx
=(x-1) - (1/2)(x-1)^2+....+ [(-1)^(n-1)/n](x-1)^n +.....

追问

说错了，应该是第五题，而且我想用间接法求

追答

consider
f(x) = 1/x^2 =>f(3) = 1/3^2
f'(x) =-2/x^3 =>f'(3)/1! = -2/3^3
f''(x) =6/x^4 =>f''(3)/2! = 3/3^4
...
f^(n)(x) = (-1)^n . (n+1)!/x^(n+2) =>f^(n)(3)/n! = (-1)^n .n/3^(n+2)
1/x^2
=1/3^2 -(2/3^3)(x-3)^2+...+[(-1)^n .n/3^(n+2)](x-3)^n +......

追问

我还想问下为什么我那么写是错的

追答

1/x^2 =1/(t+3)^2

f(t) =1/(t+3)^2 =>f(0) = 1/3^2
t=0， 展开

追问

没看懂，是我展开错了吗？

追答

那是因为不再在x=3 展开， 而是在t=0展开
f(t) =1/(t+3)^2 =>f(0) = 1/3^2

f'(t) = -2/(t+3)^3 =>f'(0)/1! = -2/3^3
...