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**HallsofIvy** Is this related to your earlier question about

$\displaystyle \left(1+\frac{1}{n}\right)^{nh}= 1+h+\frac{(nh)(nh-1)}{2!}\;\frac{1}{n^2}+.....$

for n= 1 and h= 1/2?

The left side is equal to $\displaystyle \sqrt{2}$. The right hand side is certainly NOT "3/2".

The first four partial sums are 1, 1.5, 1.375, and 1.4375. Since that is an alternating sum with decreasing terms, its sum lies between any two consecutive partial sums. In particular, the sum lies between 1.375 and 1.4375. $\displaystyle \sqrt{2}$, approximately 1.414, lies between those two, 3/2= 1.5 does not.