Question

在数列{a_n}中，已知a₁=1，a_n=n²［1+++…+］（n∈N^*，且n≥2）.

（1）当n∈N^**，n≥2时，求证：；

（2）比较(1+)(1+)…(1+)与4的大小.

Answer 1

(1)证明:当n≥2时，=1++…+，=1++…++,?

∴-=.                                                                                              ?

(2)解:当n=1时,1+=2＜4,又a₂=2²=4,                                                                   ?

当n≥2时,由(1)可知=,?

即=,                                                                                                ?

∴当n≥2时,(1+)（1+）（1+）…（）?

=···…·

=····…··（1+a_n）??

=2····…··(1+a_n)=(1+a_n)??

=+·n²·［1+++…+］.                                                            ?

∵＜=-(n≥2),?

∴1++…+＜1+(1-)+…+(-)=2-.?

∴(1+)(1+)(1+)…(1+)＜+2(2-)=4+-=4-＜4.

综上,可知(1+)(1+)…(1+)＜4.