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.

Read more about this topic: Binomial Series

Famous quotes containing the words elementary and/or bounds:

“If men as individuals surrender to the call of their elementary instincts, avoiding pain and seeking satisfaction only for their own selves, the result for them all taken together must be a state of insecurity, of fear, and of promiscuous misery.”
—Albert Einstein (1879–1955)

“What comes over a man, is it soul or mind
That to no limits and bounds he can stay confined?
You would say his ambition was to extend the reach
Clear to the Arctic of every living kind.
Why is his nature forever so hard to teach
That though there is no fixed line between wrong and right,
There are roughly zones whose laws must be obeyed?”
—Robert Frost (1874–1963)