先求和
Sn=1/2+3/2的平方+……+2n-1/2的n次方 ①
1/2Sn=1/2²+3/2³+.+(2n-3)/2ⁿ+(2n-1)/2^(n+1)②
①-②:
1/2Sn=1/2+2(1/4+1/8+.+1/2ⁿ)-(2n-1)/2^(n+1)
=1/2+2*1/4[1-1/2^(n-1)]/(1-1/2)-(2n-1)/2^(n+1)
=3/2-(2n+3)/2^(n+1)
Sn=3-(2n+3)/2ⁿ
∴lim （1/2+3/2的平方+……+2n-1/2的n次方）
=lim(n-->∞)[3-(2n+3)/2ⁿ]
=3
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