1 1.1 Excel Excel Excel log 1, log, log,, log e.7188188 ln log 1. 5cm 1mm 1 0.1mm 0.1 4 4 1 4.1 fx) fx) n0 f n) 0) x n n! n + 1 R n+1 x) fx) f0) + f 0) 1! x + f 0)! x + + f n) 0) x n + R n+1 x) n! 1
. n a n S n a k lim n a n+1 a n < 1. a n n a n a n+1 n1 a n 0 n a n S n a k n N n a n+1 a n < 0 a n > a n+1 n n+1 S a k S n a k S n+1 a k ln1 + x) fx) ln1 + x) f n) x) f n) x) 1) n 1 n 1)!1 + x) n x 0 f n) 0) 1) n 1 n 1)! ln1 + x) n 1) k 1 k 1)! x k + R n+1 x) k! n 1) k 1 x k + R n+1 x) k
a n 1)n 1 x n n lim a n+1 1) n n a n lim n 1) n 1 n lim n n n + 1 x x n+1 xn+1 1 < x < 1 n R n+1 x) ln1 + x) 1) k 1 x k k 1) k 1 x k k kn+1 x k k kn+1 1 n + 1 kn+1 x k 1 n + 1 x n+1 1 x x x n
4 ln 1 + x 1 x ln 1 + x 1 x ln1 + x) ln1 x) n 1) k 1 x k + R n+1 x) k n 1) k 1 x) k + T n+1 x) k n 1 k xk + T n+1 x) ln 1 + x 1 x ln1 + x) ln1 x) n ) 1) k 1 n x k + R n+1 x) k n 1 k 1 xk 1 + R n+1 x) T n+1 x) n n ln 1 + x n 1 x 1 k 1 xk 1 + U n+1 x) 1 b n n 1 xn 1 lim b n+1 1 b n lim n+1 xn+1 n 1 n 1 xn 1 lim n 1 n n + 1 x n x ) 1 k xk + T n+1 x) 1 < x < 1 4
U n+1 x) ln 1 + x n 1 x 1 k 1 xk 1 1 k 1 xk 1 kn+1 n + 1 kn+1 n + 1 x n+1 1 x x k 1 x 5 x x 1 ) ln 1 + x 1 x 1 < x < 1 1 < 1 + x 1 x < log 1, log, log,, log log 1 0 log log 1 1 1 1 log 4 log 00 1 + 4 ) 00 log 00 + log + ln )) 1 + 4 00 ln 1 + 4 00 )) 5
log log 1 log 9 1 log 1 1 ) 1 log + log 1 1 )) )) 1 log 4 log log 5 1 log 1 + ln 1 1 ln log 6 log + log log 7 log 7 1 log 49 1 log 1 1 ) 1 log + log 1 1 )) 1 log + log 5 + log 1 1 )) 1 log + ln )) 1 1 ln log 8 log log 9 log log 1 ln 1 + 4 ), ln 1 1 ), ln 1 1 ), ln 00 6
ln ln ln 1 ln 00 1 ln 4 1 4 ) 4 1 ln 4 + ln 00 ) 4 1 ln ln 4 ) 00 1 ln ln 1 + 4 )) 00 ln 1 + 4 ) ln 00 ln 1 + 4 ), ln 1 1 ), ln 1 1 ) ln1 + x) 00 ln ln 1 + x 1 x 6 ln 1 + 4 00 6.1 ) ln1 + x) x 4 00 R n+1 x) 1 n + 1 x n+1 1 x 1 n + 1 0.04 n+1 1 0.04 7
n R n+1 x) 1 + 1 0.04 +1 1 0.04 1 0.04 1 0.04 1 1.84 5 0.976 4.7 6 < 0.0000048 6. n ln 1 + 4 ) 4 00 00 1 ) 4 00 5.76 4 0.04 0.071 6. 0 n ) n δ 0 δ 0.0000048 0.07 ln 1 + 4 ) 0.07168 00 8
7 ln 1 1 ) 7.1 ln1 + x) x 1 R n+1 x) 1 n + 1 x n+1 1 x 1 n + 1 0.1 n+1 1 0.1 n R n+1 x) 1 + 1 0.1 +1 1 0.1 1 4 0.1 4 1 0.1 1 4 1 4 0.9 5 9 5 7. n ln 1 1 ) 1 1 ) 1 1 ) 1 0.1 0.005 0.001 0.5 0.001 9
7. n δ 5 9 5 δ 0 8 ln 0.56 ln 1 1 ) 8.1 1 1 ) 0.5 ln1 + x) x 1 R n+1 x) 1 n + 1 x n+1 1 x 1 n + 1 0.0 n+1 1 0.0 n R n+1 x) 1 + 1 0.0 +1 1 0.0 1 0.0 1 0.0 1 670.7 6 < 0.000008
8. n ln 1 1 ) 1 1 1 0.0 0.000 0.00 ) 8. n δ 0.000008 δ 0 0.0008 ln 1 1 ) 0.00000 9 ln 9.1 1 + x 1 x x 1 ln 1 + x 1 x x 1 U n+1 x) n + 1 x n+1 1 x n + 1 1 n+1 ) 1 1 11
n 4 U n+1 x) 4 + 1 9 1 ) 9 8 9) 1 787 1.7 5 < 0.00001 1 4+1 ) 1 1 9. n 4 1 ln 1 1 + 1 56 76545 0.6914 ) 1 + 1 5 ) 5 1 + 1 7 ) ) 7 1 9. n 4 δ 0 δ 0.00001 0.6914 ln 0.69148 1
ln ln ln ln 1 + 4 ) 00 [0.6914, 0.69148] + [ 0.07168, 0.07] [6.907680, 6.90776] ln [.0546,.05878] [.054,.059] ln 11 log log 1 + ln )) 1 + 4 00 ln 0. + [ [0.07, 0.07168] [.054,.059] 0. + 0.07 0.07168, 0. +.059.054 [0.097, 0.001] log ] 1 log log 1 1 + ln 1 1 ln )) [ 0.56, 0.5] 0.5 + [.054,.059] [ 0.5 0.56 ] 0.5, 0.5.054.059 [0.477, 0.47718] 1
log 1 log 7 log 7 1 log + ln 1 1 ln )) 1 1 [ 0.0008, 0.00000] log + [.054,.059] 1 1 [ ] 0.00000 log.059, 0.0008.054 1 1 [0.097, 0.001] [0.00486, 0.004871] [ 1 0.001 0.004870, 1 0.097 ] 0.004864 [0.849785, 0.849885] [0.84978, 0.84989] log 14 log 1 0 log [0.097, 0.001] log [0.477, 0.47718] log 4 [0.600594, 0.60060] log 5 [0.6989699, 0.698970] log 6 [0.778149, 0.778159] log 7 [0.84978, 0.84989] log 8 [0.900891, 0.90090] log 9 [0.95440, 0.95456] log 1 0 4 7 8 6 8 7 4 7 1 5 1.1 6 1. 6 1.6 5 0 14
15,4 ln n 4 ln [0, 1. 5 ] log 0. 1 4 log, log 7 n log n 15