简算1+3/2+5/4+7/8+9/16+...+4003/2^2001
3个回答
S = 1+3/2+5/4+7/8+9/16+...+4003/2^2001
两端同时乘 1/2
S/2 = 1/2 + 3/4 + 5/8 + 7/16 + 9/32 + …… + 4001/2^2001 + 4003/2^2002
两式相减
S - S/2
= 1 + (3/2 - 1/2) + (5/4 - 3/4) + (7/8 - 5/8) + …… + (4003/2^2001 -4001/2^2001) - 4003/2^2002
S/2 = 1 + 2*(1/2 + 1/4 + 1/8 + …… + 1/2^2001) - 4003/2^2002
= 1 + 2 * (1/2) * [1 - (1/2)^2001]/(1 -1/2) - 4003/2^2002
= 1 + 2 * ( 1 - 1/2^2001) - 4003/2^2002
= 3 - 1/2^2000 - 4003/2^2002
= 3 - 1/2^2000 * ( 1 + 4003/2^2)
= 3 - 4007/2^2002
S = 6 - 4007/2^2001
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