Exercise 26 in Schutz's First course in GR

[MathematicalPhysicist]

Homework Statement

The question as follows:

Calculate the energy that is required to accelerate a particle of rest mass $m\ne 0$ from speed $v$ to speed $v+\delta v$ ($\delta v \ll v$).
Show that it would take an infinite amount of energy to accelerate the particle to the speed of light.

Homework Equations

The Attempt at a Solution

Here's what I have done so far, the 4-momentum before is $(m,mv)$ and the 4-momentum after is: $(E,m(v+\delta v))$, the square of the 4 momentum is conserved, i.e:
$$E^2 - m^2(v+\delta v)^2 = m^2-m^2v^2$$

After rearranging I get the following equation for the energy:

$$E=mv\sqrt{1/v^2+2\delta v /v +(\delta v/v)^2}$$

I think I have a mistake somewhere, since I don't know how to expand this in a Taylor series, I have the expansion $\sqrt{1+x} \approx 1+1/2 x$, but here I have $1/v^2$ inside the sqrt.

Perhaps I am wrong with the 4-momentum or something else, any tips?

Thanks in advance.

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[MathematicalPhysicist]

I think I got it wrong it should be $E=\gamma(v+\delta v)mc^2$, and $$\gamma(v+\delta v) \approx 1+1/2 (v+\delta v)^2$$

For ##v +\delta v = c$, we get the E diverges.