1. Taylor series

Recommended reference: Strang [Str23], in particular Section 10.4.

Taylor series are a useful way of approximating a function by simpler functions, namely polynomials.

1.1. Definition and basic properties

Recall that a power series is like a polynomial, but with possibly infinitely many powers of \(x\):

\[ g(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots \]

where the \(c_i\) are scalars (usually real or complex numbers).

Definition 1.1 (Taylor series)

Let \(f\) be a (real or complex) function that is infinitely differentiable at a point \(a\). The Taylor series of \(f\) at \(a\) is the power series

\[ f(a) + f'(a) (x - a) + \frac{f''(a)}{2}(x - a)^2 + \cdots = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n. \]

For almost all functions \(f\) that one encounters in practice, the Taylor series of \(f\) will converge to \(f\). This means that the polynomials

\[ f(a) + f'(a) (x - a) + \frac{f''(a)}{2}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n \]

of degree \(n\) obtained by truncating the Taylor series to a finite number of terms give successively better approximations to \(f\) as \(n\to\infty\), at least for \(x\) sufficiently close to \(a\). However, there are exceptions; see the exercises.

1.2. Important examples

1.2.1. The exponential function

Consider the function \(f(x)=\exp(x)\). To compute its Taylor series at \(x=0\), note that all derivatives of \(f\) are equal to \(f\) itself, and take the value \(1\) at \(x=0\). This gives

\[\begin{split} \begin{aligned} \exp(x) &= 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3+\cdots\\ &= \sum_{n=0}^\infty\frac{1}{n!}x^n. \end{aligned} \end{split}\]

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from myst_nb import glue
import numpy as np
import plotly.graph_objects as go
import sympy as sp
from sympy.abc import x

from _modules.series_utils import get_pointwise_values

# Define the function and domain
f = sp.exp(x)
domain = np.arange(-1, 1, 0.01)

# Create figure
fig = go.Figure()

# Add the original function trace (f(x) = e^x)
fig.add_trace(
    go.Scatter(
        visible=True,
        line=dict(color="red", width=2),
        x=domain,
        y=np.exp(domain),
        showlegend=True,
        name="f(x) = exp(x)"
    )
)

# Add Taylor series approximation traces
for num_terms in range(1, 11):
    series = sp.series(f, x, 0, num_terms)
    series_trunc = series.removeO()
    
    fig.add_trace(
        go.Scatter(
            visible=False,
            line=dict(color="blue", width=2),
            x=domain,
            y=get_pointwise_values(series_trunc, domain),
            showlegend=True,
            name=f"{num_terms}-term approx."
        )
    )

# Make the last Taylor series approximation trace (10th term) visible by default
fig.data[2].visible = True

# Create and add slider
steps = []
for i in range(1, len(fig.data)):
    # Ensure that the original function trace is always visible
    step = dict(
        method="update",
        args=[{"visible": [True] + [False] * (len(fig.data) - 1)}],
        label=f"{i}"
    )
    step["args"][0]["visible"][i] = True  # Toggle i-th Taylor series trace to "visible"
    steps.append(step)

sliders = [dict(
    active=2,  # Set the 10-term approximation as the default
    currentvalue={"prefix": "Num. of terms: "},
    pad={"t": 50},
    steps=steps
)]

fig.update_layout(
    sliders=sliders,
    height=600,
    width=750,
    title="Taylor series approximation of f(x) = exp(x) around x = 0",
    xaxis_title="x",
    yaxis_title="f(x)"
)

glue("exponential", fig)

Fig. 1.1 An exponential function and an approximation to its Taylor series around \(x=0\).

1.2.2. The logarithm

Now consider the function \(f(x)=\log(1-x)\). We have

\[ f'(x) = -\frac{1}{1-x} \]

and more generally

\[ f^{(n)}(x) = -\frac{(n-1)!}{(1-x)^n}\quad\text{for all }n\ge1. \]

This gives

\[\begin{split} \begin{aligned} \log(1-x) &= -x-\frac{1}{2}x^2-\frac{1}{3}x^3-\cdots\\ &= -\sum_{n=1}^\infty\frac{1}{n}x^n. \end{aligned} \end{split}\]

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from myst_nb import glue
import numpy as np
import plotly.graph_objects as go
import sympy as sp
from sympy.abc import x

# Define the function and domain
f = sp.log(1 - x)
domain = np.arange(-1, 1, 0.01)

# Create figure
fig = go.Figure()

# Add the original function trace (f(x) = log(1 - x))
fig.add_trace(
    go.Scatter(
        visible=True,
        line=dict(color="red", width=2),
        x=domain,
        y=np.log(1 - domain),
        showlegend=True,
        name="f(x) = log(1 - x)"
    )
)

# Add Taylor series approximation traces
for num_terms in range(1, 11):
    series = sp.series(f, x, 0, num_terms)
    series_trunc = series.removeO()
    
    fig.add_trace(
        go.Scatter(
            visible=False,
            line=dict(color="blue", width=2),
            x=domain,
            y=get_pointwise_values(series_trunc, domain),
            showlegend=True,
            name=f"{num_terms}-term approx."
        )
    )

# Make the last Taylor series approximation trace (10th term) visible by default
fig.data[2].visible = True

# Create and add slider
steps = []
for i in range(1, len(fig.data)):
    # Ensure that the original function trace is always visible
    step = dict(
        method="update",
        args=[{"visible": [True] + [False] * (len(fig.data) - 1)}],
        label=f"{i}"
    )
    step["args"][0]["visible"][i] = True  # Toggle i-th Taylor series trace to "visible"
    steps.append(step)

sliders = [dict(
    active=2,  # Set the 10-term approximation as the default
    currentvalue={"prefix": "Num. of terms: "},
    pad={"t": 50},
    steps=steps
)]

fig.update_layout(
    sliders=sliders,
    height=600,
    width=750,
    title="Taylor series approximation of f(x) = log(1 - x) around x = 0",
    xaxis_title="x",
    yaxis_title="f(x)"
)

glue("logarithmic", fig)

Fig. 1.2 A logarithmic function and an approximation to its Taylor series around \(x=0\).

1.2.3. Trigonometric functions

The Taylor series of the cosine and sine functions are given by

\[\begin{split} \begin{aligned} \cos x &= \sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n},\\ \sin x &= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}. \end{aligned} \end{split}\]

This can be proved directly using the familiar expressions for the derivatives of these functions. It is also instructive to derive the above formulae by taking the Taylor series of \(\exp(ix)\), splitting it into the real and imaginary parts and using Euler’s formula

\[ \exp(ix) = \cos x + i\sin x. \]

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1.3. Exercises

Exercise 1.1

Compute the Taylor series at \(x=0\) of the function

\[ f(x) = \frac{1}{1-x}. \]

Exercise 1.2

Derive the Taylor series for the sine and cosine functions using the two different methods sketched in Trigonometric functions.

Exercise 1.3

Do one or more of exercises 25–44 from Strang [Str23], Chapter 10, where you are asked to find the first few terms of the Taylor series for various functions.

Exercise 1.4

Consider the function

\[\begin{split} f(x) = \begin{cases} \exp(-1/x^2)& \text{if }x\ne0,\\ 0& \text{if }x=0. \end{cases} \end{split}\]

Show that the Taylor series of \(f\) at \(0\) is identically zero.