某等差数列,lim(n接近无限大)Sn/(an)²=1/6,求公差.

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某等差数列,lim(n接近无限大)Sn/(an)²=1/6,求公差.
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某等差数列,lim(n接近无限大)Sn/(an)²=1/6,求公差.
某等差数列,lim(n接近无限大)Sn/(an)²=1/6,求公差.

某等差数列,lim(n接近无限大)Sn/(an)²=1/6,求公差.
设公差为d,则
an=a1+(n-1)d=dn+a1-d
Sn=na1+n(n-1)d/2
∴Sn/a²n=[na1+n(n-1)d/2]/(dn+a1-d)²
∴lim(n-->∞)Sn/(an)²
=lim(n-->∞)[na1+n(n-1)d/2]/(dn+a1-d)²
=lim(n-->∞)[na1+n(n-1)d/2]/(d²n²+2d(a1-d)n+(a1-d)²]
=lim(n-->∞)[a1/n+(1-1/n)d/2]/(d²+2d(a1-d)/n+(a1-d)²/n²]
=(d/2)/d²=1/(2d)
∵lim(n接近无限大)Sn/(an)²=1/6
∴1/(2d)=1/6 ∴d=3

3

你把,登差的设出来

Sn=(a1+an)*n/2=na1+n(n-1)d/2
an=a1+(n-1)d
Sn/(an)^2=(na1+n(n-1)d/2)/(a1+(n-1)d)^2
=(na1+n^2d/2-nd/2)/(a1^2+2(n-1)a1d+(n-1)^2d^2) 上下除以n^2
=(d/2+a1/n-d/2n)/((1-1/n)^2*d^2+2(n-1)a1d/n^2+a1^2/n^2)
=(d/2)/(d^2) (n→无穷)
=1/2d
所以d=3

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