Sequence summation 1 / 3 + 1 / 5 + 1 / 7 + +1/21=?

Sequence summation 1 / 3 + 1 / 5 + 1 / 7 + +1/21=?

The general formula is: 1 / (2n + 1)
Because:
1+1/2+1/3+…… +1/(2n+1)=ln(2n+1)+0.5772157.(1)
1+1/2+1/3+…… +1 / N = lnn + 0.5772157 deformation, both sides divide by 2 at the same time
1/2+1/4+…… +1/2n=1/2*lnn+0.5772157/2...(2)
(1) - (2) get
1+1/3+…… +1/(2n+1)=ln(2n+1)-1/2*lnn+0.5772157/2
So: 1 / 3 + +1/(2n+1)=ln(2n+1)-1/2*lnn-0.71138215
When 1 / (2n + 1) = 1 / 21, n = 10
Substitute n = 10 into the above formula
Get 1 / 3 + 1 / 5 + 1 / 7 + +1/21=1n21-0.5ln10-0.71138215=1.18184774122 .
However, because the problem n is relatively small, there is error in using the formula, because the result calculated by the calculator is 1.18087457777
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