求一道高数题4.2.1
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由绝对收敛数列:1-x+x^2-...+(-1)^n x^n, n->oo = 1/(1+x), n from 0 to oo
两边积分 from 0 to x: x - x^2/2 + ... + (-1)^(n+1) x^(n)/n + .. = ln(1+x), n from 1 to oo
由此可得:
ln(a+x)
= ln(a(1+x/a))
= lna + ln(1+x/a)
= lna + x/a - (x/a)^2/2 + ... + (-1)^(n+1) (x/a)^n/n, n from 1 to oo
两边积分 from 0 to x: x - x^2/2 + ... + (-1)^(n+1) x^(n)/n + .. = ln(1+x), n from 1 to oo
由此可得:
ln(a+x)
= ln(a(1+x/a))
= lna + ln(1+x/a)
= lna + x/a - (x/a)^2/2 + ... + (-1)^(n+1) (x/a)^n/n, n from 1 to oo
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