.. _chap-more-taylor-series:

********************
 More taylor series
********************

.. _sec-starting-to-study-exponentials:

Exponentials
============


The number :math:`e`: base for natural exponentials and logarithms
------------------------------------------------------------------

In earlier lessons we discussed the exponential function at length.
Here we discuss some of the reasons for which we consider Euler's
number :math:`e` to be the "natural" base for exponentials and
logarithms, analogously to how radians are the natural unit of measure
for circles.

Part of this is a visual exploration of :math:`e^x` and an graphical
look at its slope, to then conclude that:

.. math::

   \frac{d e^x}{dx} = e^x

which is not true for other bases, like :math:`2^x` and :math:`10^x`

We can also mention that:

.. math::

   e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n

and calculate it for values of n like 1, 10, 100, 1000, 10000, ...  We
can also calculate :math:`e` with:

.. math::

   e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} +
   \frac{1}{4!} + \frac{1}{5!} + \dots = \sum_{k=0}^{\infty}
   \frac{1}{k!} \approx 2.71828182846

but the justification for that will only come when we show how to
calculate the Taylor series for :math:`e^x`.


.. caution::

   It is also worth mentioning that difference communities in math,
   science, and engineering use :math:`\log(x)` to mean the *natural*
   logarithm (base :math:`e`).  Other communities will use
   :math:`\log(x)` to mean the *base 10* logarithm, and they say
   :math:`\ln(x)` for the natural logarithm.  I use the former
   approach, and the various programming language libraries (C,
   Python, ...) do the same.



But we take a break from these notes as we get to use a proper text
book to explore exponentials and logarithms in more detail.


The Taylor series for :math:`e^x` -- visually
---------------------------------------------

First we look at it visually.  We will want to plot terms in the
following way in gnuplot and geogebra and desmos:

.. code-block:: console

   $ gnuplot
   ## then the following lines have the prompt gnuplot> and we type:
   reset
   set grid
   set xrange [-1:3]
   set terminal qt linewidth 3
   plot exp(x) lw 2
   replot 1
   replot 1 + x
   replot 1 + x + x**2 / 2!
   replot 1 + x + x**2 / 2! + x**3 / 3!
   replot 1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4!
   replot 1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5!
   replot 1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5! + x**6 / 6!

In geogebra or desmos:

::

   e^x
   1
   1 + x
   1 + x + x**2 / 2!
   1 + x + x**2 / 2! + x**3 / 3!
   1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4!
   1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5!
   1 + x + x**2 / 2! + x**3 / 3! + x**4 / 4! + x**5 / 5! + x**6 / 6!

While looking at the visual output I discuss with the students how the
polynomial approximations look different to the left and to the right
of the origin: to the right the polynomials all approximate from
below, while to the left the even/odd alternating polynomials
approximate from above/below respectively.

The Taylor series for :math:`e^x` -- calculating the coefficients
-----------------------------------------------------------------

Remembering that :math:`\frac{de^x}{dx} = e^x` we can calculate:

.. math::

   \frac{d^k e^x}{dx^k} \bigg \rvert_{x = 0} = e^0 = 1

which gives us Taylor coefficients :math:`a_k = \frac{1}{k!}` and the
series looks like:

.. math::

   e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!}
   + \frac{x^3}{3!} + \frac{x^4}{4!} + \dots

Here I usually spend some time discussing how the :math:`k!` on the
denominator makes this series converge, even when :math:`x^k` can get
really big.  Later I will contrast that with other denominators and we
should make sure we are always that the series will converge.

I also often take off on another tangent here: I write out the series
for :math:`e^x` and underneath it I write the series for
:math:`\cos(x)` and :math:`\sin(x)`.  In doing this I *align* all the
terms so it looks like this:

.. math::
   :nowrap:

   \begin{align*}
   e^x =& \; 1 & + x & + \frac{x^2}{2!}
   & + \frac{x^3}{3!} & + \frac{x^4}{4!} & + \frac{x^5}{5!} & + \dots
   \\
   \cos(x) =& \; 1 & {} & - \frac{x^2}{2!}
   & {} & + \frac{x^4}{4!} &{} & + \dots
   \\
   \sin(x) =& \; {} & x & {} &
   - \frac{x^3}{3!} & {} & + \frac{x^5}{5!} & + \dots
   \end{align*}

I then have them speculate on whether it looks like the exponential
series might contain the :math:`sin()` and :math:`cos()` series, but
with funny alternations of + and - signs.

It's hard to get that alternation, but I show that if you take the
series for :math:`e^{ix}` then you get the series for :math:`\cos(x) +
i\sin(x)`.  This is known as :index:`Euler's identity`.

An interesting result of Euler's identity is that:

.. math::

   e^{i \pi} = -1 \rightarrow e^{i \pi} + 1 = 0

a remarkable identity that combines those remarkable numbers :math:`0,
1, e, i, {\rm and} \; \pi`.

In talking about this I do the following:

#. Remind students in passing that the way we treat :math:`i` in
   complex numbers is like any other letter (like :math:`a, x, w,
   \dots`) except that when we see :math:`i^2` we replace it with -1.
   This means that:

   .. math::

      i^0 = 0\\
      i^1 = i\\
      i^2 = -1\\
      i^3 = -i\\
      i^4 = 1\\
      i^5 = i

   which gives us *exactly* the alternation of negatives and :math:`i`
   that we need to spot Euler's identity in the Taylor series for
   exponential, sin, and cos.
#. I point out how weird it is that a *non-oscillating* function like
   the exponential seems to contain sin and cos.  I point out that
   when you enter the complex plane things change quite a bit, and
   :math:`e^{ix}` is definitely an oscillating function.
#. In this vein I point out that when they study electronics they will
   see that :math:`e^{i\omega t}` makes a frequent appearance in the
   calculation of even the most elementary electrical circuits.

Miscellaneous Taylor expansions
===============================

.. rubric:: Logarithms:

The first thing to note is that we cannot expand the Taylor series for
the logarithm around :math:`x = 0` because it has a singularity there.

We could either introduce the general Taylor series centered about
:math:`f(a)`, but we have decided to not introduce that level of
detail in this working group, so we look at :math:`\log(1-x)` and
:math:`\log(1+x)` which get around that problem.

.. math::

   log(1 - x) = -\sum_{k=1}^{\infty} \frac{x^k}{k} \\
   log(1 + x) = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k}

the first when :math:`|x| < 1`, the second when :math:`-1 < x \leq 1`.
It is very important to point out these restrictions on :math:`x`.

Since students are often just learning calculus, it is useful to work
out the details of (say) :math:`\log(1-x)`.  Remembering that
:math:`\frac{d\log(x)}{dx} = 1/x` and :math:`\frac{dx^n}{dx} = n\times
x^{n-1}` (even when :math:`n` is negative!):

.. math::
   :nowrap:

   \begin{align*}
   0: &= \log(1-0) = 0 \\
     &\rightarrow \fbox{a_0 = 0}
     \\
   1: &= \frac{d\log(1-x)}{dx} \bigg \rvert_{x=0}
     = \frac{1}{1-x} \frac{d(1-x)}{dx} \bigg \rvert_{x=0}
     = \frac{-1}{1-x} \bigg \rvert_{x=0} = -1
     \\
     &\rightarrow \fbox{a_1 = -1}
     \\
   2: &= \frac{d^2\log(1-x)}{dx^2} \bigg \rvert_{x=0}
     = \frac{d\frac{-1}{(1-x)}}{dx} \bigg \rvert_{x=0}
     = \frac{-1 \times (-1) \times (-1)}{(1-x)^2} \bigg \rvert_{x=0} = -1
     \\
     &\rightarrow \fbox{a_2 = -1/2! = -1/2}
     \\
   3: &= \frac{d^3\log(1-x)}{dx^3} \bigg \rvert_{x=0}
     = \frac{d\frac{-1}{(1-x)^2}}{dx} \bigg \rvert_{x=0}
     = \frac{-1 \times (-2) \times (-1)}{(1-x)^3} \bigg \rvert_{x=0} = -2 \times 1
     \\
     &\rightarrow \fbox{$a_1 = -2! / 3! = -1/3$}
     \\
   4: &= \frac{d^4\log(1-x)}{dx^4} \bigg \rvert_{x=0}
     = \frac{d\frac{-2\times 1}{(1-x)^3}}{dx} \bigg \rvert_{x=0}
     = \frac{- (-3) \times 2 \times (-1)}{(1-x)^4} \bigg \rvert_{x=0}
     = -3 \times 2 \times 1 = -3!
     \\
     &\rightarrow \fbox{$a_1 = -3! / 4! = -1/4$}
     \\
   k: &= \frac{d^k\log(1-x)}{dx^k} \bigg \rvert_{x=0}
     = \frac{d\frac{-k\times\dots\times 1}{(1-x)^3}}{dx} \bigg \rvert_{x=0}
     = \frac{- (-(k-1)) \times \dots \times (-1)}{(1-x)^4} \bigg \rvert_{x=0}
     = -(k-1)!
     \\
     &\rightarrow \fbox{$a_k = -(k-1)! / k! = -1/k$}
   \end{align*}

Note that when you plot these logarithmic functions you will need to
double check that your plotting program uses :math:`log()` for
*natural* logarithms.  Some plotting programs (like Desmos) use
:math:`\ln()` for natural logarithms.

.. rubric:: Geometric series:

.. math::
   :nowrap:

   \begin{eqnarray}
   \frac{1}{1-x}      = & \sum_{k=0}^{\infty} & x^k \\
   \frac{1}{(1-x)^2}  = & \sum_{k=1}^{\infty} & k x^{k-1} \\
   \frac{1}{(1-x)^3}  = & \sum_{k=2}^{\infty} & \frac{(k-1)k}{2} x^{k-2}
   \end{eqnarray}

when :math:`|x| < 1`


Some square root expansions
===========================

Square root functions can get complicated.  For example, the
relativistic formula for the rest *plus* kinetic energy of an object
with mass :math:`m_0` is

.. math::

   E_{\rm total} = \frac{m_0 c^2}{\sqrt{1 - \frac{v^2}{c^2}}}

This has the famous Lorenz gamma factor:

.. math::

   \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}

We sometimes use a shorthand :math:`\beta = v/c`, where :math:`\beta`
is the velocity expressed as a *fraction of the speed of light*, and
get:

.. math::

   \gamma = \frac{1}{\sqrt{1 - \beta^2}}


The first few terms in the taylor series expansion in :math:`\beta`
are (see the Cupcake Physics link in the resources chapter for
details):

..
   .. math::

      \gamma \;\; = \; & 1 & + \frac{1}{2}\beta^2         & + & \frac{3}{8}\beta^4         & + & \frac{5}{16}\beta^6         & + & \frac{35}{128}\beta^8 \dots\\
                  = \; & 1 & + \frac{1}{2}\frac{v^2}{c^2} & + & \frac{3}{8}\frac{v^4}{c^4} & + & \frac{5}{16}\frac{v^6}{c^6} & + & \frac{35}{128}\frac{v^8}{c^8}\dots

.. math::
   :nowrap:

   \begin{eqnarray}
   \gamma \;\; = \; & 1  + \frac{1}{2}\beta^2          +  \frac{3}{8}\beta^4          +  \frac{5}{16}\beta^6          +  \frac{35}{128}\beta^8 \dots\\
               = \; & 1  + \frac{1}{2}\frac{v^2}{c^2}  +  \frac{3}{8}\frac{v^4}{c^4}  +  \frac{5}{16}\frac{v^6}{c^6}  + & \frac{35}{128}\frac{v^8}{c^8}\dots
    \end{eqnarray}

Putting this back into the formula for energy we get:

.. math::

   E_\textrm{total} = \; \frac{m c^2}{\sqrt{1 - \frac{v^2}{c^2}}}
                 = \; m c^2 + \frac{1}{2} m v^2 + \dots

For low values of :math:`v^2/c^2` (i.e. :math:`v` much slower than the
speed of light) we have:

.. math::

   E_{\rm total} = m c^2 + \frac{1}{2} m v^2 + \dots

We can read off the terms and realize that the total energy is equal
to the famous rest mass :math:`E_{\rm rest} = m c^2` plus the
kinetic energy :math:`\frac{1}{2} m v^2 + \dots`:

.. math::

   E_{\rm total} = E_{\rm rest} + E_{\rm kinetic} = m c^2 +
   \frac{1}{2} m v^2
   + \frac{3}{8} m \frac{v^4}{c^2} \dots

Let us explore the Lorenz gamma factor for values of :math:`v` in the
whole range from 0 to :math:`c`:

::

   $ gnuplot
   ## then the following lines have the prompt gnuplot> and we type:
   reset
   set grid
   set ylabel '\gamma'
   set xlabel '\beta (v/c)'
   set xrange [0:1]
   set terminal qt linewidth 3
   plot 1 / sqrt(1 - x**2)

Or in a web-based graphing calculator:

::

   1 / (1 - x^2)^(1/2)

.. _fig-lorenz_factor:

.. figure:: lorenz_factor.*
   :width: 60%

   The lorenz factor as a function of :math:`\beta = v/c`.  Note
   how it is close to 1 for most of the run, but grows out of control
   when :math:`v` approaches the speed of light :math:`c`.


What insight does this give us on the energy of an object as it
approaches the speed of light?  Note that the formulae for length and
time are:

.. math::

   L = \frac{1}{\gamma} L_0 \\
   \Delta t' = \gamma \Delta t

so the behavior of :math:`\gamma` as a function of :math:`\beta` (and
thus :math:`v`) also affects length and time.

Now let us look at the polynomial approximates in :math:`\beta`:

::

   $ gnuplot
   ## then the following lines have the prompt gnuplot> and we type:
   reset
   set grid
   set ylabel '\gamma'
   set xlabel '\beta (v^2/c^2)'
   set xrange [0:0.0001]
   set terminal qt linewidth 3
   plot 1 / (1 - x**2)
   replot 1
   replot 1 + (1.0/2) * x**2
   replot 1 + (1.0/2) * x**2 + (3.0/8) * x**4
   replot 1 + (1.0/2) * x**2 + (3.0/8) * x**4 + (5.0/16) * x**6