Binomial Series - Elementary Bounds On The Coefficients

Elementary Bounds On The Coefficients

In order to keep the whole discussion within elementary methods, one may derive the asymptotics (5) proving the inequality

with

as follows. By the inequality of arithmetic and geometric means

$\left|{\alpha \choose k} \right|^2=\prod_{j=1}^k \left|1-\frac{1+\alpha}{j}\right|^2 \leq \left( \frac{1}{k}\sum_{j=1}^{k} \left|1-\frac{1+\alpha}{j}\right|^2 \right)^k.$

Using the expansion

the latter arithmetic mean writes

$\frac{1}{k}\sum_{j=1}^{k} \left|1-\frac{1+\alpha}{j}\right|^2= 1+\frac{1}{k}\left(- 2(1+\mathrm{Re}\,\alpha) \sum_{j=1}^{k}\frac{1}{j}+|1+\alpha|^2\sum_{j=1}^{k}\frac{1}{j^2}\right)\ .$

To estimate its kth power we then use the inequality

that holds true for any real number r as soon as 1 + r/k ≥ 0. Moreover, we have elementary bounds for the sums:

Thus,

with

proving the claim.

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