第三小题求解。谢谢。
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(3)
an = n^2
To prove : Sn = 1/a1+1/a2+...+1/an < 7/4
n=1 , S1 =1/1 =1 < 7/4
n=2 , S2 =1+ 1/4 =5/4 < 7/4
n=3 , S3 =5/4 + 1/9= 49/36 < 7/4
n=4 , S4 =49/36 + 1/16= 205/144 < 7/4
for n>=5
Sn
=S4 + (1/a5+1/a6+...+1/an)
=205/144 + (1/5^2 +1/6^2+...+1/n^2)
<205/144 + { 1/(4x5) +1/(5x6)+...+1/[(n-1)n] }
=205/144 + ( 1/4 -1/n )
< 205/144 +1/4
=241/144
< 7/4
an = n^2
To prove : Sn = 1/a1+1/a2+...+1/an < 7/4
n=1 , S1 =1/1 =1 < 7/4
n=2 , S2 =1+ 1/4 =5/4 < 7/4
n=3 , S3 =5/4 + 1/9= 49/36 < 7/4
n=4 , S4 =49/36 + 1/16= 205/144 < 7/4
for n>=5
Sn
=S4 + (1/a5+1/a6+...+1/an)
=205/144 + (1/5^2 +1/6^2+...+1/n^2)
<205/144 + { 1/(4x5) +1/(5x6)+...+1/[(n-1)n] }
=205/144 + ( 1/4 -1/n )
< 205/144 +1/4
=241/144
< 7/4
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