inequality anyone?
- From: Darren <anon5874@xxxxxxxxx>
- Date: 18 May 2007 14:31:34 -0700
Need help with proving this inequality (to do with proving that a
power-series is analytic):
|\frac{z^{k+1} - b^{k+1}}{z-b} - (k+1)b^k |
\le \frac{(|b| + |z-b|)^{k+1} - |b|^{k+1}}{|z-b|} - (k+1)|b|^k
\le (k+1)[(|b| + |z-b|)^k - |b|^k]
I start with |z| \le |b| + |z-b|, and get the first term of the second
line.
But why subtract the term "(k+1)|b|^k" ? Wouldn't you add for a
triangle inequality?
And, how do you go from the 2nd line to the 3rd line?
The 3rd line can be re-written:
(k+1)[(|b| + |z-b|)^k - |b|^k] = k(|b| + |z-b|)^k + (|b| + |z-b|)^k -
(k+1)|b|^k
The 2nd and 3rd term on the r.h.s can easily be gotten, but I am
having trouble getting "k(|b| + |z-b|)^k".
Thanks!
.
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