Taylor Series Demonstration

This notebook explores Taylor polynomial approximations and their error behavior.

import numpy as np
import matplotlib.pyplot as plt
from math import factorial

Taylor Polynomials of $e^x$ ¶

The Taylor series for $e^x$ centered at $x_0 = 0$ is:

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

(1)

The $n$ -th degree Taylor polynomial $P_n(x)$ truncates this series.

def taylor_exp(x, n):
    """Compute n-th degree Taylor polynomial of e^x at x=0."""
    return sum(x**k / factorial(k) for k in range(n + 1))

# Vectorize for array input
taylor_exp_vec = np.vectorize(taylor_exp, excluded=['n'])

x = np.linspace(-2, 3, 200)

fig, ax = plt.subplots(figsize=(10, 6))

# Plot exact function
ax.plot(x, np.exp(x), 'k--', linewidth=2, label=r'$e^x$ (exact)')

# Plot Taylor polynomials
colors = plt.cm.viridis(np.linspace(0.2, 0.8, 4))
for i, n in enumerate([1, 2, 3, 4]):
    y_approx = taylor_exp_vec(x, n)
    error_at_1 = abs(taylor_exp(1, n) - np.e)
    ax.plot(x, y_approx, color=colors[i], linewidth=1.5,
            label=f'$P_{n}(x)$, error at $x=1$: {error_at_1:.2e}')

ax.set_xlim(-2, 3)
ax.set_ylim(-1, 10)
ax.set_xlabel('$x$', fontsize=12)
ax.set_ylabel('$y$', fontsize=12)
ax.set_title('Taylor Polynomial Approximations of $e^x$', fontsize=14)
ax.legend(loc='upper left', fontsize=10)
ax.grid(True, alpha=0.3)
ax.axhline(y=0, color='k', linewidth=0.5)
ax.axvline(x=0, color='k', linewidth=0.5)

plt.tight_layout()
plt.show()

Taylor Polynomials of $\tan(x)$ ¶

Let’s approximate $\tan(x)$ near $x_0 = 0$ . The Taylor series is:

\tan(x) = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots

(2)

We’ll examine how the remainder term bounds the true error.

def taylor_tan(x, n):
    """Taylor polynomial of tan(x) at x=0.
    
    Only odd terms are nonzero. We compute coefficients numerically.
    """
    # Coefficients for tan(x): 1, 0, 1/3, 0, 2/15, 0, 17/315, ...
    coeffs = [0, 1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835]
    result = 0
    for k in range(min(n + 1, len(coeffs))):
        result += coeffs[k] * x**k
    return result

taylor_tan_vec = np.vectorize(taylor_tan, excluded=['n'])

x = np.linspace(0, 0.5, 100)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Function and approximations
ax1.plot(x, np.tan(x), 'k-', linewidth=2, label=r'$\tan(x)$ (exact)')
for n, color in [(1, 'C0'), (3, 'C1'), (5, 'C2')]:
    ax1.plot(x, taylor_tan_vec(x, n), '--', color=color, linewidth=1.5, label=f'$P_{n}(x)$')

ax1.set_xlabel('$x$', fontsize=12)
ax1.set_ylabel('$y$', fontsize=12)
ax1.set_title(r'Taylor Approximations of $\tan(x)$', fontsize=14)
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)

# Right: Absolute errors
for n, color in [(1, 'C0'), (3, 'C1'), (5, 'C2')]:
    error = np.abs(np.tan(x) - taylor_tan_vec(x, n))
    ax2.semilogy(x, error + 1e-16, '-', color=color, linewidth=1.5, label=f'$|\\tan(x) - P_{n}(x)|$')

ax2.set_xlabel('$x$', fontsize=12)
ax2.set_ylabel('Absolute Error', fontsize=12)
ax2.set_title('Approximation Error', fontsize=14)
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

The Remainder Term¶

Taylor’s theorem tells us that for $P_n(x)$ centered at $x_0$ :

f(x) = P_n(x) + R_n(x)

(3)

where the remainder is:

R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - x_0)^{n+1}

(4)

for some $\xi$ between $x_0$ and $x$ . This gives us an error bound.

# For e^x, all derivatives are e^x, so |R_n(x)| <= e^|x| * |x|^(n+1) / (n+1)!

x_val = 0.5

print(f"Approximating e^{x_val} = {np.exp(x_val):.10f}")
print()
print(f"{'n':>3} | {'P_n(x)':>14} | {'True Error':>12} | {'Bound |R_n|':>12} | {'Bound ok?':>10}")
print("-" * 65)

for n in range(1, 8):
    approx = taylor_exp(x_val, n)
    true_error = abs(np.exp(x_val) - approx)
    # Upper bound: max of e^x on [0, 0.5] is e^0.5
    bound = np.exp(x_val) * x_val**(n+1) / factorial(n+1)
    ok = "Yes" if true_error <= bound else "No"
    print(f"{n:3d} | {approx:14.10f} | {true_error:12.2e} | {bound:12.2e} | {ok:>10}")

Key Observations¶

Taylor polynomials provide increasingly accurate approximations near the expansion point
The remainder term gives a rigorous error bound that always overestimates the true error
Convergence is local: Taylor polynomials work best near $x_0$ ; accuracy degrades as you move away
Higher degree = higher accuracy near $x_0$ , but may oscillate wildly far from $x_0$

Taylor Polynomials of exe^xex¶

Taylor Polynomials of tan⁡(x)\tan(x)tan(x)¶

The Remainder Term¶

Key Observations¶

Taylor Polynomials of $e^x$ ¶

Taylor Polynomials of $\tan(x)$ ¶