```
In [1]:
```%matplotlib inline
import numpy as np
import scipy as sc
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import sympy as sp
import functools
import itertools
sns.set();

Let $f$ be a function with all derivatives throughout some interval containing $a$ as an interior point. Then the Taylor series generated by $f$ at $x = a$ is given by the following power series

$$f(x) = \sum_{k = 0}^{\infty} \frac{f^{k}(a)}{k!} (x-a)^{k}$$where $k!$ denotes the factorial of $k$ and $f^{k}(a)$ denotes the nth derivative of $f$ evaluated at the point $a$.

Taking $a = 0$, we have the McLaurin series:

$$f(x) = \sum_{k = 0}^{\infty} \frac{f^{k}(0)}{k!} x^{k}$$Let $f$ be a function with derivatives of order $k$ for $k = 1, 2, \dots, N$ in some interval containing $a$ as an interior point. Then for any integer $n$ from 0 through $N$, the Taylor polynomial of order $n$ generated by $f$ at the point $x = a$ is the polynomial

$$P_{n}(x) = \sum_{k = 0}^{n} \frac{f^{k}(a)}{k!} (x-a)^{k}$$where $k!$ denotes the factorial of $k$ and $f^{k}(a)$ denotes the nth derivative of $f$ evaluated at the point $a$.

If $f$ and its first $n$ derivatives $f', f'', \dots, f^{n}$ are continuous on the closed interval between $a$ and $b$, and $f^{n}$ is differentiable on the open interval between $a$ and $b$, then there exists a number $c$ between $a$ and $b$ such that

$$f(b) = \sum_{k = 0}^{n} \frac{f^{k}(a)}{k!} (x-a)^{k} + R_{n}(x)$$where

$$R_{n}(x) = \frac{f^{n+1}(c)}{(n+1)!} (x-a)^{n+1}$$for some $c$ between $a$ and $x$. The function $R_{n}(x)$ is called the Lagrange remainder.

In other words,

$$f(x) = P_{n}(x) + R_{n}(x)$$where $P_{n}(x)$ is the Taylor polynomial seen above.

We can write our own function of Taylor series as below.

```
In [2]:
```def taylor(f, x, var, max_terms=6, x0=0):
def taylor_terms():
for k in range(max_terms):
term = (sp.diff(f, var, k).subs(var, x0).evalf()/np.math.factorial(k)) * (x - x0)**k
yield term
serie = 0
for term in taylor_terms():
serie += term
return serie

Consider the function $f:\mathbb{R}\rightarrow[-1,1]$ defined as $f(x) = \cos(x)$.

Its Taylor expansion is given by

$$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n)!}x^{2n}, \forall x \in \mathbb{R}$$The code below show approximates for this function with 3, 10, 15 and 20 terms in the summation (or orders of the Taylor polynomials). The actual value for the function is also computed.

```
In [3]:
```x = sp.symbols("x")
f = sp.cos(x)
f_series = functools.partial(taylor, var=x)
df1 = pd.DataFrame({"x": np.linspace(-8, 8, num=100)})
max_terms = [3, 10, 15, 20]
for max_term in max_terms:
df1["y" + str(max_term)] = np.array([f_series(f, x, max_terms=max_term) for x in df1["x"]])
df1["y_actual"] = np.cos(df1["x"])
df1.head()

```
Out[3]:
```

```
In [4]:
```fig, ax = plt.subplots(figsize=(13, 8))
ax.set_ylim(-1.5, 1.5)
colors = ["red", "purple", "blue", "green"]
for max_term, color in zip(max_terms, colors):
plt.plot(df1["x"], df1["y"+str(max_term)], color=color)
plt.plot(df1["x"], df1["y_actual"], color='black')
plt.legend(loc='lower left', fancybox=True, framealpha=1, shadow=True, borderpad=1, frameon=True)
ax.set(title=r"Taylor polynomials for $\cos(x)$", xlabel="x", ylabel="y");

```
```

Consider the function $f:\mathbb{R}\rightarrow[-1,1]$ defined as $f(x) = \exp(x)$.

Its Taylor expansion is given by

$$\exp(x) = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}, \forall x \in \mathbb{R}$$The code below show approximates for this function with 4, 7, 10 and 12 terms in the summation (or orders of the Taylor polynomials). The actual value for the function is also computed.

```
In [5]:
```x = sp.symbols("x")
f = sp.exp(x)
f_series = functools.partial(taylor, var=x)
df2 = pd.DataFrame({"x": np.linspace(0, 10, num=100)})
max_terms = [4, 7, 10, 12]
for max_term in max_terms:
df2["y" + str(max_term)] = np.array([f_series(f, x, max_terms=max_term) for x in df2["x"]])
df2["y_actual"] = np.exp(df2["x"])
df2.head()

```
Out[5]:
```

```
In [6]:
```fig, ax = plt.subplots(figsize=(13, 8))
ax.set_ylim(0, 1000)
colors = ["red", "purple", "blue", "green"]
for max_term, color in zip(max_terms, colors):
plt.plot(df2["x"], df2["y"+str(max_term)], color=color)
plt.plot(df2["x"], df2["y_actual"], color='black')
plt.legend(loc='lower left', fancybox=True, framealpha=1, shadow=True, borderpad=1, frameon=True)
ax.set(title=r"Taylor polynomials for $\exp(x)$", xlabel="x", ylabel="y");

```
```