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Infinite Sequences and Series

1. Infinite Sequence and Series

Definition

An infinite sequence is an ordered list of elements, typically numbers, that extends indefinitely. Each term in the sequence is usually defined by a formula.

A series is the sum of the terms of a sequence. An infinite series is the sum of an infinite sequence:

\[ S = a_1 + a_2 + a_3 + \dots + a_n + \dots \]

Explanation and Intuition

  • Infinite Sequence: Think of a sequence as an endless list of numbers arranged in a specific order. For example, \( \{1, 1/2, 1/3, 1/4, \dots\} \) is an infinite sequence where each term is \( 1/n \).
  • Infinite Series: When we add the terms of a sequence, we form a series. For instance, the series for \( \{1, 1/2, 1/3, \dots\} \) is:
\[ S = 1 + \frac{1}{2} + \frac{1}{3} + \dots \]

2. Convergence Test of Infinite Series

Convergence and Divergence

A series converges if the sum of its terms approaches a finite value as the number of terms goes to infinity. Otherwise, it diverges.

Tests for Convergence

a) Geometric Series Test

A geometric series has the form:

\[ S = a + ar + ar^2 + ar^3 + \dots \]
  • Converges if \( |r| < 1 \).
  • Diverges if \( |r| \geq 1 \).

Example (Python Code):

import numpy as np
import matplotlib.pyplot as plt

# Geometric series example
def geometric_series(a, r, n):
    terms = [a * (r ** i) for i in range(n)]
    return terms

# Parameters
a, r, n = 1, 0.5, 20
terms = geometric_series(a, r, n)
cumsum = np.cumsum(terms)

plt.plot(cumsum, marker='o', label='Cumulative Sum')
plt.axhline(y=sum(terms), color='r', linestyle='--', label='Convergent Value')
plt.xlabel('Number of Terms')
plt.ylabel('Sum')
plt.title('Convergence of Geometric Series')
plt.legend()
plt.show()

b) p-Series Test

A p-series has the form:

\[ S = \sum_{n=1}^{\infty} \frac{1}{n^p} \]
  • Converges if \( p > 1 \).
  • Diverges if \( p \leq 1 \).

Example (Python Code):

# p-series example
def p_series(p, n):
    terms = [1 / (i ** p) for i in range(1, n + 1)]
    return terms

# Parameters
p, n = 2, 50
terms = p_series(p, n)
cumsum = np.cumsum(terms)

plt.plot(cumsum, marker='o', label=f'p = {p}')
plt.xlabel('Number of Terms')
plt.ylabel('Sum')
plt.title('Convergence of p-Series')
plt.legend()
plt.show()

c) Ratio Test

For a series \( \sum a_n \), if:

\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
  • Converges if \( L < 1 \).
  • Diverges if \( L > 1 \).
  • Inconclusive if \( L = 1 \).

Example:

The factorial series \( \sum \frac{1}{n!} \) converges because the ratio test gives \( L = 0 \).

3. Power Series, Taylor, and Maclaurin Series

Power Series

A power series is a series of the form:

\[ S(x) = \sum_{n=0}^{\infty} c_n (x - a)^n \]
  • \( a \): Center of the series.
  • \( c_n \): Coefficients.

Example (Python Code):

# Power series example
x = np.linspace(-2, 2, 100)
a, c = 0, [1, -1/2, 1/3, -1/4]

# Compute power series
def power_series(x, c, a):
    return sum(c[i] * (x - a) ** i for i in range(len(c)))

y = power_series(x, c, a)

plt.plot(x, y, label='Power Series')
plt.axhline(0, color='k', linewidth=0.5)
plt.axvline(0, color='k', linewidth=0.5)
plt.xlabel('x')
plt.ylabel('S(x)')
plt.title('Visualization of Power Series')
plt.legend()
plt.show()

Taylor and Maclaurin Series

  • Taylor Series: Expands a function \( f(x) \) around a point \( a \):
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \]
  • Maclaurin Series: Special case of the Taylor series where \( a = 0 \).

Example (Maclaurin Series for \( e^x \)):

# Maclaurin series for e^x
from math import factorial

def maclaurin_exponential(x, terms):
    return sum((x ** n) / factorial(n) for n in range(terms))

x_vals = np.linspace(-2, 2, 100)
y_vals = [maclaurin_exponential(x, 10) for x in x_vals]

plt.plot(x_vals, np.exp(x_vals), label='e^x (Exact)')
plt.plot(x_vals, y_vals, '--', label='Maclaurin Series (Approximation)')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Maclaurin Series Approximation for e^x')
plt.legend()
plt.show()