Given sequence Sn = 2An + (- 1) ^ n n is greater than one, try to prove that for any m greater than 4, 1 / A4 + 1 / A5 + 1 / A6 +. + 1 / am is less than 7 / 8 I calculate an = [2 ^ (n-1) - 2 (- 1) ^ n] / 3

Given sequence Sn = 2An + (- 1) ^ n n is greater than one, try to prove that for any m greater than 4, 1 / A4 + 1 / A5 + 1 / A6 +. + 1 / am is less than 7 / 8 I calculate an = [2 ^ (n-1) - 2 (- 1) ^ n] / 3

An=[2^(n-1)-2(-1)^n]/3 =(2/3)[2^(n-2)+(-1)^(n-1)].
A4=2.
If M is even, then
1/A4 +1/A5 +1/A6 +...+1/Am
=1/A4 +(1/A5+1/A6)+...+[1/A(m-1)+1/Am]
Where 1 / a (m-1) + 1 / am
＝(3/2){1/[2^(m-3)+1]+1/[2^(m-2)-1]}
＝(3/2){[2^(m-3)+1+2^(m-2)-1]/[2^(m-3)+1][2^(m-2)-1]}
＝(3/2){[2^(m-3)+2^(m-2)]/[2^(2m-5)-2^(m-3)+2^(m-2)-1]
＜(3/2){[2^(m-3)+2^(m-2)]/[2^(2m-5)][∵-2^(m-3)+2^(m-2)-1＞0]
=(3/2)[1/2^(m-3)+1/2^(m-2)]
∴1/A4 +1/A5 +1/A6 +...+1/Am
=1/A4 +(1/A5+1/A6)+...+(1/A(m-1)+1/Am)
＜(1/2)+(3/2)[1/2^3+1/2^4+1/2^5+...+1/2^(m-2)]
＝(1/2)+(3/2)(1/4)[1-1/2^(m-4)]
＜(1/2)+(3/8)
=7/8.
When m is odd, M + 1 is even
1/A4 +1/A5 +1/A6 +...+1/Am
＜1/A4 +1/A5 +1/A6 +...+1/Am+1/A(m+1)
＜7/8.
To sum up, the proposition is proved