前n项求和 an=arctan(1/（2n^2))

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前n项求和 an=arctan(1/（2n^2))
答案是 arctan(2n+1)-pi/4 这个答案又是怎么出来的

数学归纳法
S(1) = a(1) = arctan(1/2),
S(2) = a(1) + a(2) = arctan(1/2) + arctan(1/8),
tan[S(2)] = (1/2 + 1/8)/[1 - (1/2)(1/8)] = 10/15 = 2/3,
S(2) = arctan(2/3),
S(3) = S(2) + a(3) = arctan(2/3) + arctan[1/18]
tan[S(3)] = (2/3+1/18)/[1-(2/3)(1/18)] = 13*3/(26*2) = 3/4.
S(3) = arctan(3/4).
推测 S(n) = arctan[n/(n+1)]
=arctan [2n/(2n+2)]
=arctan {[(2n+1)-1]/[(2n+1)+1]}
=arctan {[tan arctan(2n+1) -tan π/4]/[1+ tan arctan(2n+1) • tan π/4]}
=arctan {tan [arctan(2n+1) - π/4]}
=arctan(2n+1) - π/4.
假设当n=k时,
Sn=arctan(2k+1)-π/4成立,则
tan S(k)=[tan arctan(2k+1) -tan π/4]/[1+tan arctan(2k+1) • tan π/4]
=[tan arctan(2k+1) - 1]/[tan arctan(2k+1) + 1]
=[(2k+1) - 1]/[(2k+1) + 1]
=2k/（2k+2）=k/（k+1）
则当n=k+1时,
tan S(k+1)=[tan S(k) + tan a（k+1）]/[1- tan S(k) • tan a（k+1）]
=[k/（k+1）+1/2（k+1）^2]/[1-(k/（k+1）)•(1/2（k+1）^2)]
=[2k（k+1）^2+（k+1）]/[2（k+1）^3-k]
=[（k+1）(2k^2+2k+1)]/[2k^3+6k^2+5k+2]
=[（k+1）(2k^2+2k+1)]/[(2k^2+2k+1)(k+2)]
=（k+1）/（k+2）
即当n=k+1时,tan S(k+1)=（k+1）/[（k+1）+1]成立.
即S(k+1)=arctan （k+1）/[（k+1）+1]
=arctan [2(k+1) +1]- π/4成立.
则说明Sn=arctan(2n+1)-π/4成立.

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