![]() |
December 15th, 2017, 10:23 AM | #1 |
Senior Member Joined: Jan 2015 From: usa Posts: 100 Thanks: 0 | Taylor's series
Let $\nu\ge 1$ be a parameter. For all $t>0,$ we consider \begin{align} A(t) & =\frac{1-\cos(t\sqrt{4\nu-1})}{4\nu-1}-(\cosh(t)-1) \\[10pt] g(t) & =\frac{\frac{\sin(t\sqrt{4\nu-1})}{\sqrt{4\nu-1}}+\sinh(t)}{A(t)} \end{align} By using Taylor's series, I want to prove that there exists a constant $c>0$ which doesn't depend on $\nu$ such that $$\frac{\nu t^3}{\ln\left(1-\frac{2}{g(t)+1}\right)}\le c$$ for all $t\le \frac{1}{\sqrt{4\nu-1}}$ Can you please help me to do so. Thanks. |
![]() |
December 15th, 2017, 12:59 PM | #2 |
Global Moderator Joined: May 2007 Posts: 6,496 Thanks: 579 |
First step: Let $\displaystyle T=\frac{1}{\sqrt{4\nu -1}}\ so\ that\ \nu=\frac{1}{4}(1+\frac{1}{T^2})$. The algebra should be easier.
|
![]() |
December 15th, 2017, 01:09 PM | #3 |
Senior Member Joined: Jan 2015 From: usa Posts: 100 Thanks: 0 |
@mathman, This is what i wrote: for all $t\le \frac{1}{\sqrt{c}}$ $$sin(t\sqrt{c})=t\sqrt{c}-\frac{(t\sqrt{c})^3}{3!}+..+(-1)^n\frac{(t\sqrt{c})^{2n+1}}{(2n+1)!}+O((t\sqrt{c })^{2n+3})$$ for all $t\le \frac{1}{\sqrt{c}}$ for all $t\le \frac{1}{\sqrt{c}}$ $$cos(t\sqrt{c})=1-\frac{(t\sqrt{c})^2}{2!}+..+(-1)^n\frac{(t\sqrt{c})^{2n}}{(2n)!}+O((t\sqrt{c})^{ 2n+1})$$ $$\cosh(t)=1+\frac{t^2}{2!}+..+\frac{t^{2n}}{(2n)! }+O(t^{2n+1})$$ $$\sinh(t)=1+\frac{t^3}{3!}+..+\frac{t^{2n+1}}{(2n +1)!}+O(t^{2n+3})$$ But i don't know how to find the result. |
![]() |
December 16th, 2017, 01:33 PM | #4 |
Global Moderator Joined: May 2007 Posts: 6,496 Thanks: 579 |
Study the behavior of $\displaystyle 1-\frac{2}{g(t)+1}$. For ln to be > 0 for all t, the expressions must be > 1 for all t.
|
![]() |
![]() |
|
Tags |
series, taylor |
Thread Tools | |
Display Modes | |
|
![]() | ||||
Thread | Thread Starter | Forum | Replies | Last Post |
Taylor Series, am I doing right? | WWRtelescoping | Complex Analysis | 3 | April 7th, 2014 02:58 AM |
In need of help disk, series test, taylor, and power series | g0bearmon | Real Analysis | 2 | May 22nd, 2012 12:10 PM |
Taylor Series | stewpert | Calculus | 3 | February 10th, 2010 01:15 PM |
Taylor Series of log(1-t) | vjj | Number Theory | 0 | December 31st, 1969 04:00 PM |
In need of help disk, series test, taylor, and power series | g0bearmon | Calculus | 1 | December 31st, 1969 04:00 PM |