Confused on what a series means. I have been reading the book Relativity: The special & General Theory where in chapter XV, the author develops the expression of kinetic energy
$$
\frac{1}{2} mv^2
$$
or
$$
\text{ }m\frac{v^2}{2}
$$
in the form of a series,
$$
mc^2+\text{ }m\frac{v^2}{2}+\frac{3}{8}m\frac{v^4}{c^2}+...
$$
Can someone explain to me what a series is? Or at the very least show me where I can find more information about this? Thank you.
 A: With respect to the original formula,and
usually $v<<c$
$$E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}=\\mc^2(1-\frac{v^2}{c^2})^{-\frac12}\\
\sim mc^2 (1-(-\frac12)\frac{v^2}{c^2}+...)\\=mc^2+\frac 12mv^2+...$$
A: As the other answer is rather formal, let me provide my own short one, dealing with the problem at hand.
The exact expression for the relativistic kinetic energy is:
$$K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2$$
That is, total energy minus the rest energy.
In the non-relativistic limit we have the following assumption:
$$v^2 \ll c^2$$
Or:
$$\beta=\frac{v^2}{c^2} \ll 1$$
Turns out that there's a useful sequence of approximations that converge to the square root, provided that $\beta$ is small in absolute value as indicated above.
$$\sqrt{1+\beta}=1+\frac{\beta}{2}-\frac{\beta^2}{8}+\cdots$$
Where $\cdots$ should be replaced by as many terms as it takes to get the desired accuracy.
This is a case of so called Taylor series, and I encourage the OP to look it up.
In this particular case, it's enough to take only the first term, which gives us:
$$K \approx mc^2 \left( \frac{1}{1-v^2/(2c^2)}-1 \right)$$
But this is still too complicated.
However, we know another Taylor series, or rather a simple case of geometric progression sum:
$$\frac{1}{1-\beta} = 1+\beta+\beta^2+\cdots$$
Again, it's enough to use the first term, which gives us:
$$K \approx mc^2 \cdot \frac{v^2}{2c^2}=\frac{mv^2}{2}$$
For what it's worth, if you don't want to get too formal, a convergent series is a sequence of sums with increasing number of terms that gives a good approximation to some finite value. Usually, we can even estimate the error of the approximation.
Example:
$$\frac{\pi}{4}=1-\frac13+\frac15-\frac17+\cdots$$

*

*the famous Leibniz series for Pi.

A good estimation for the error is half the last term.
A: I'll explain what a series is based on my studies.
A series of general term $ a_{n} $ denoted $ \sum\limits_{n\geq 0}{a_{n}} $ is a mathematical object, precisely the couple formed by the sequence $ \left(a_{n}\right)_{n\geq 0} $ and the sequence of partial sums $ \left(\sum\limits_{k=0}^{n}{a_{k}}\right)_{n\geq 0} $ : $$ \left(\left(a_{n}\right)_{n\geq 0},\left(\sum\limits_{k=0}^{n}{a_{k}}\right)_{n\geq 0}\right) $$
Whenever the sequence $ \left(\sum\limits_{k=0}^{n}{a_{k}}\right)_{n\geq 0} $ converges to some $ \ell $, we say that the series $ \sum\limits_{n\geq 0}{a_{n}} $ is convergent, we call $ \ell $ the sum of the series, and we write : $$ \sum_{n=0}^{+\infty}{a_{n}}=\ell $$
If the sequence $ \left(\sum\limits_{k=0}^{n}{a_{k}}\right)_{n\geq 0} $ diverges, we say that the series $ \sum\limits_{n\geq 0}{a_{n}} $ is divergent.
This is basically the formal definition of a series.
