How can I expand this expression in powers of 1/c²?

[Joao Victor]

Homework Statement

I was studying James B. Hartle's book - Gravity: an introduction to Einstein's General Relativity -, and in one section he expanded an expression in powers of 1/c², but I couldn't follow what he did. I do know this is related to a Taylor Series Expansion, and I do know how to construct the taylor expansion for a function f(x) [or even for a function of several variables], but I have no idea on how to proceed with this kind of expansion presented on the book.

Relevant Equations

e = \frac {mc^2+E} {mc^2} = 1 + \frac {2E} {mc^2} + ...

As I said before, I really have no idea on how to proceed.

 

[HallsofIvy]

You almost have it! $$\frac{mc^2+ E}{mc^2}= 1+ \frac{E}{mc^2}= 1+ \frac{E}{m}\frac{1}{c^2}$$. That is the 'power series" $$a_0+ a_1\frac{1}{c^2}+ a_2\left(\frac{1}{c^2}\right)^2+ \cdot\cdot\cdot$$ with $$a_0= 1$$, $$a_1= \frac{E}{m}$$, and all other coefficients equal to 0.

 

[vela]

Problem Statement: I was studying James B. Hartle's book - Gravity: an introduction to Einstein's General Relativity -, and in one section he expanded an expression in powers of 1/c², but I couldn't follow what he did. I do know this is related to a Taylor Series Expansion, and I do know how to construct the taylor expansion for a function f(x) [or even for a function of several variables], but I have no idea on how to proceed with this kind of expansion presented on the book.
Relevant Equations: e = \frac {mc^2+E} {mc^2} = 1 + \frac {2E} {mc^2} + ...

As I said before, I really have no idea on how to proceed.

The total energy per unit rest mass, $e$, is defined as
$$e = \frac {mc^2+E_\text{Newt}} {mc^2},$$ but in equation (9.53), the lefthand side is $e^2$. That's why the two appears in the second term. It's not a Taylor series expansion. Just multiply it out.