Sequences and Series
Disclaimer: These are my personal notes compiled for my own reference and learning. They may contain errors, incomplete information, or personal interpretations. While I strive for accuracy, these notes are not peer-reviewed and should not be considered authoritative sources. Please consult official textbooks, research papers, or other reliable sources for academic or professional purposes.
1. Sequences
1.1 Definition
A sequence is a function $f: \mathbb{N} \rightarrow \mathbb{R}$, usually denoted as $\{a_n\}_{n=1}^{\infty}$ or simply $\{a_n\}$.
The $n$-th term is $a_n = f(n)$.
1.2 Convergence of Sequences
Definition: A sequence $\{a_n\}$ converges to $L$ if:
$$\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } \forall n \geq N: |a_n - L| < \epsilon$$
We write $\lim_{n \to \infty} a_n = L$ or $a_n \to L$.
1.3 Properties of Convergent Sequences
- Uniqueness: If $a_n \to L$ and $a_n \to M$, then $L = M$
- Boundedness: Every convergent sequence is bounded
- Algebra of Limits: If $a_n \to L$ and $b_n \to M$, then:
- $a_n + b_n \to L + M$
- $a_n \cdot b_n \to L \cdot M$
- $\frac{a_n}{b_n} \to \frac{L}{M}$ (if $M \neq 0$)
- Squeeze Theorem: If $a_n \leq b_n \leq c_n$ and $a_n \to L$, $c_n \to L$, then $b_n \to L$
1.4 Monotone Sequences
Monotone Convergence Theorem: Every bounded monotone sequence converges.
- Increasing: $a_n \leq a_{n+1}$ for all $n$
- Decreasing: $a_n \geq a_{n+1}$ for all $n$
1.5 Subsequences
A subsequence $\{a_{n_k}\}$ is obtained by selecting terms at indices $n_1 < n_2 < n_3 < \ldots$
Bolzano-Weierstrass Theorem: Every bounded sequence has a convergent subsequence.
1.6 Cauchy Sequences
Definition: A sequence $\{a_n\}$ is Cauchy if:
$$\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } \forall m,n \geq N: |a_m - a_n| < \epsilon$$
Cauchy Criterion: A sequence converges if and only if it is Cauchy.
2. Series
2.1 Definition
Given a sequence $\{a_n\}$, the series $\sum_{n=1}^{\infty} a_n$ is defined as the limit of partial sums:
$$S_N = \sum_{n=1}^{N} a_n$$
The series converges if $\lim_{N \to \infty} S_N$ exists and is finite.
2.2 Convergence Tests
2.2.1 Divergence Test
If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges.
Note: If $\lim_{n \to \infty} a_n = 0$, the series may converge or diverge.
2.2.2 Integral Test
If $f(x)$ is positive, continuous, and decreasing for $x \geq 1$, then:
$$\sum_{n=1}^{\infty} f(n) \text{ and } \int_1^{\infty} f(x) dx$$
either both converge or both diverge.
2.2.3 Comparison Tests
Direct Comparison: If $0 \leq a_n \leq b_n$ for all $n$:
- If $\sum b_n$ converges, then $\sum a_n$ converges
- If $\sum a_n$ diverges, then $\sum b_n$ diverges
Limit Comparison: If $a_n, b_n > 0$ and $\lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, then $\sum a_n$ and $\sum b_n$ have the same convergence behavior.
2.2.4 Ratio Test
For $\sum a_n$ with $a_n > 0$, let $L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}$:
- If $L < 1$, the series converges
- If $L > 1$, the series diverges
- If $L = 1$, the test is inconclusive
2.2.5 Root Test
For $\sum a_n$ with $a_n \geq 0$, let $L = \lim_{n \to \infty} \sqrt[n]{a_n}$:
- If $L < 1$, the series converges
- If $L > 1$, the series diverges
- If $L = 1$, the test is inconclusive
2.3 Alternating Series
Alternating Series Test (Leibniz): If $\{a_n\}$ is decreasing and $a_n \to 0$, then $\sum (-1)^n a_n$ converges.
Error Estimation: For alternating series, the error in using $S_N$ as an approximation is at most $|a_{N+1}|$.
2.4 Absolute and Conditional Convergence
- Absolutely Convergent: $\sum |a_n|$ converges
- Conditionally Convergent: $\sum a_n$ converges but $\sum |a_n|$ diverges
Theorem: If $\sum |a_n|$ converges, then $\sum a_n$ converges.
3. Power Series
3.1 Definition
A power series centered at $a$ is:
$$\sum_{n=0}^{\infty} c_n (x-a)^n$$
where $c_n$ are coefficients and $a$ is the center.
3.2 Radius of Convergence
For every power series, there exists $R \geq 0$ (possibly $\infty$) such that:
- The series converges for $|x-a| < R$
- The series diverges for $|x-a| > R$
- Convergence at $|x-a| = R$ must be checked separately
Formula for Radius of Convergence:
$$R = \frac{1}{\lim_{n \to \infty} \left|\frac{c_{n+1}}{c_n}\right|} \quad \text{or} \quad R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|c_n|}}$$
3.3 Properties of Power Series
- Term-by-term differentiation: Inside the radius of convergence
- Term-by-term integration: Inside the radius of convergence
- Uniform convergence: On compact subsets of the interval of convergence
4. Taylor and Maclaurin Series
4.1 Taylor Series
If $f$ is infinitely differentiable at $a$, the Taylor series is:
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$
4.2 Maclaurin Series
Taylor series centered at $a = 0$:
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$
4.3 Common Maclaurin Series
- $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$
- $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$
- $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots$
- $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \ldots$ for $|x| < 1$
- $(1+x)^r = \sum_{n=0}^{\infty} \binom{r}{n} x^n$ for $|x| < 1$
- $\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots$ for $|x| < 1$
4.4 Taylor's Theorem with Remainder
Lagrange Form:
$$f(x) = \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n + \frac{f^{(N+1)}(c)}{(N+1)!}(x-a)^{N+1}$$
where $c$ is between $a$ and $x$.
5. Special Series
5.1 Geometric Series
$$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{for } |r| < 1$$
5.2 Harmonic Series
$$\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots \quad \text{(diverges)}$$
5.3 p-Series
$$\sum_{n=1}^{\infty} \frac{1}{n^p} \begin{cases}
\text{converges} & \text{if } p > 1 \\
\text{diverges} & \text{if } p \leq 1
\end{cases}$$
5.4 Telescoping Series
Series where consecutive terms cancel:
$$\sum_{n=1}^{\infty} (a_n - a_{n+1}) = a_1 - \lim_{n \to \infty} a_n$$
6. Fourier Series
6.1 Definition
For a periodic function $f(x)$ with period $2\pi$:
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos(nx) + b_n \sin(nx)\right)$$
where:
$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$$
$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$$
6.2 Convergence of Fourier Series
Dirichlet Conditions: If $f$ is periodic, piecewise continuous, and has finitely many maxima and minima in one period, then the Fourier series converges to:
- $f(x)$ at points of continuity
- $\frac{f(x^+) + f(x^-)}{2}$ at points of discontinuity
7. Code Examples
# Python implementations for sequences and series analysis
import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate
import sympy as sp
class SequenceAnalyzer:
def __init__(self, formula, symbolic_var='n'):
"""
Initialize with a formula for the sequence
formula: symbolic expression or lambda function
"""
if isinstance(formula, str):
self.symbolic_var = sp.Symbol(symbolic_var)
self.formula = sp.sympify(formula)
self.func = sp.lambdify(self.symbolic_var, self.formula, 'numpy')
else:
self.func = formula
self.formula = None
def generate_terms(self, n_terms=50, start=1):
"""Generate first n_terms of the sequence"""
indices = np.arange(start, start + n_terms)
return indices, self.func(indices)
def plot_sequence(self, n_terms=50, start=1):
"""Plot the sequence"""
indices, terms = self.generate_terms(n_terms, start)
plt.figure(figsize=(10, 6))
plt.plot(indices, terms, 'bo-', markersize=4, linewidth=1)
plt.xlabel('n')
plt.ylabel('a_n')
plt.title('Sequence Terms')
plt.grid(True, alpha=0.3)
plt.show()
def check_convergence(self, n_terms=1000, tolerance=1e-10):
"""Heuristic check for convergence"""
indices, terms = self.generate_terms(n_terms)
# Check if terms approach zero (necessary condition)
if abs(terms[-1]) > tolerance:
return False, f"Last term |a_{n_terms}| = {abs(terms[-1]):.2e} > {tolerance}"
# Check if sequence is Cauchy
tail_length = min(100, n_terms // 4)
tail = terms[-tail_length:]
max_diff = np.max(np.abs(np.diff(tail)))
if max_diff < tolerance:
return True, f"Sequence appears to converge (max difference in tail: {max_diff:.2e})"
else:
return False, f"Sequence may not converge (max difference in tail: {max_diff:.2e})"
class SeriesAnalyzer:
def __init__(self, term_formula, symbolic_var='n'):
"""
Initialize with a formula for series terms
"""
if isinstance(term_formula, str):
self.symbolic_var = sp.Symbol(symbolic_var)
self.formula = sp.sympify(term_formula)
self.func = sp.lambdify(self.symbolic_var, self.formula, 'numpy')
else:
self.func = term_formula
self.formula = None
def partial_sums(self, n_terms=100, start=1):
"""Calculate partial sums"""
indices = np.arange(start, start + n_terms)
terms = self.func(indices)
partial_sums = np.cumsum(terms)
return indices, terms, partial_sums
def ratio_test(self, n_terms=100, start=1):
"""Apply ratio test"""
indices = np.arange(start, start + n_terms)
terms = np.abs(self.func(indices))
# Calculate ratios a_{n+1}/a_n
ratios = terms[1:] / terms[:-1]
# Remove any infinite or NaN values
valid_ratios = ratios[np.isfinite(ratios)]
if len(valid_ratios) == 0:
return None, "No valid ratios found"
# Check if ratios approach a limit
tail_ratios = valid_ratios[-20:] # Last 20 ratios
limit = np.mean(tail_ratios)
if limit < 1:
return "Convergent", f"Ratio test: limit ≈ {limit:.4f} < 1"
elif limit > 1:
return "Divergent", f"Ratio test: limit ≈ {limit:.4f} > 1"
else:
return "Inconclusive", f"Ratio test: limit ≈ {limit:.4f} = 1"
def plot_convergence(self, n_terms=100, start=1):
"""Plot partial sums to visualize convergence"""
indices, terms, partial_sums = self.partial_sums(n_terms, start)
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 8))
# Plot terms
ax1.plot(indices, terms, 'bo-', markersize=3, linewidth=1)
ax1.set_xlabel('n')
ax1.set_ylabel('a_n')
ax1.set_title('Series Terms')
ax1.grid(True, alpha=0.3)
# Plot partial sums
ax2.plot(indices, partial_sums, 'ro-', markersize=3, linewidth=1)
ax2.set_xlabel('n')
ax2.set_ylabel('S_n')
ax2.set_title('Partial Sums')
ax2.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()
return partial_sums[-1]
class PowerSeries:
def __init__(self, coefficients, center=0):
"""
Initialize power series with coefficients
coefficients: list or function that gives c_n
center: center of the series
"""
self.coefficients = coefficients
self.center = center
def radius_of_convergence(self, n_terms=100):
"""Estimate radius of convergence using ratio test"""
if callable(self.coefficients):
coeffs = [self.coefficients(n) for n in range(n_terms)]
else:
coeffs = self.coefficients[:n_terms]
# Remove zeros to avoid division issues
nonzero_coeffs = [c for c in coeffs if c != 0]
if len(nonzero_coeffs) < 2:
return float('inf')
# Calculate ratios |c_n|/|c_{n+1}|
ratios = []
for i in range(len(nonzero_coeffs) - 1):
if nonzero_coeffs[i+1] != 0:
ratios.append(abs(nonzero_coeffs[i]) / abs(nonzero_coeffs[i+1]))
if ratios:
return np.mean(ratios[-10:]) # Average of last 10 ratios
else:
return float('inf')
def evaluate(self, x, n_terms=50):
"""Evaluate power series at point x"""
if callable(self.coefficients):
coeffs = [self.coefficients(n) for n in range(n_terms)]
else:
coeffs = self.coefficients[:n_terms]
result = 0
power = 1
for n, c in enumerate(coeffs):
result += c * power
power *= (x - self.center)
return result
def plot_function(self, x_range=(-2, 2), n_terms=50, num_points=1000):
"""Plot the function represented by the power series"""
x = np.linspace(x_range[0], x_range[1], num_points)
y = [self.evaluate(xi, n_terms) for xi in x]
plt.figure(figsize=(10, 6))
plt.plot(x, y, 'b-', linewidth=2)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title(f'Power Series (first {n_terms} terms)')
plt.grid(True, alpha=0.3)
plt.show()
def taylor_series_demo():
"""Demonstrate Taylor series for common functions"""
x = sp.Symbol('x')
# Define functions and their Taylor series
functions = {
'exp(x)': sp.exp(x),
'sin(x)': sp.sin(x),
'cos(x)': sp.cos(x),
'1/(1-x)': 1/(1-x),
'ln(1+x)': sp.log(1+x)
}
for name, func in functions.items():
print(f"\n{name}:")
# Calculate Taylor series at x=0
series = func.series(x, 0, 6).removeO()
print(f"Taylor series: {series}")
# Convert to power series coefficients
poly = sp.Poly(series, x)
coeffs = poly.all_coeffs()
coeffs.reverse() # Reverse to get [c_0, c_1, c_2, ...]
print(f"Coefficients: {coeffs}")
# Convergence tests implementation
def convergence_tests(series_analyzer, n_terms=100):
"""Apply various convergence tests"""
print("=== Convergence Analysis ===")
# Ratio test
result, explanation = series_analyzer.ratio_test(n_terms)
print(f"Ratio Test: {result}")
print(f" {explanation}")
# Check if terms approach zero (necessary condition)
_, terms, _ = series_analyzer.partial_sums(n_terms)
last_term = terms[-1]
print(f"\nDivergence Test:")
if abs(last_term) < 1e-10:
print(f" Terms approach zero (a_{n_terms} ≈ {last_term:.2e})")
else:
print(f" Terms do not approach zero (a_{n_terms} = {last_term:.4f})")
print(" Series diverges by divergence test!")
# Example usage
if __name__ == "__main__":
print("=== Sequence Analysis ===")
# Example 1: Sequence 1/n
seq1 = SequenceAnalyzer(lambda n: 1/n)
converges, msg = seq1.check_convergence()
print(f"Sequence 1/n: {msg}")
# Example 2: Sequence (-1)^n/n
seq2 = SequenceAnalyzer(lambda n: (-1)**n / n)
converges, msg = seq2.check_convergence()
print(f"Sequence (-1)^n/n: {msg}")
print("\n=== Series Analysis ===")
# Example 1: Harmonic series
harmonic = SeriesAnalyzer(lambda n: 1/n)
print("Harmonic series 1/n:")
convergence_tests(harmonic)
# Example 2: Geometric series with r = 1/2
geometric = SeriesAnalyzer(lambda n: (0.5)**n)
print("\nGeometric series (1/2)^n:")
convergence_tests(geometric)
# Example 3: Alternating harmonic series
alt_harmonic = SeriesAnalyzer(lambda n: (-1)**(n+1) / n)
print("\nAlternating harmonic series (-1)^(n+1)/n:")
convergence_tests(alt_harmonic)
print("\n=== Power Series ===")
# Geometric series 1/(1-x) = sum(x^n)
geom_coeffs = [1] * 50 # All coefficients are 1
ps = PowerSeries(geom_coeffs)
R = ps.radius_of_convergence()
print(f"Geometric series radius of convergence: {R:.4f}")
# Evaluate at x = 0.5
value = ps.evaluate(0.5)
analytical = 1 / (1 - 0.5)
print(f"Series at x=0.5: {value:.6f}")
print(f"Analytical 1/(1-0.5): {analytical:.6f}")
print("\n=== Taylor Series Demo ===")
taylor_series_demo()
8. Applications
8.1 Mathematical Analysis
- Foundation for calculus and real analysis
- Function approximation and numerical methods
- Fourier analysis and signal processing
8.2 Physics and Engineering
- Quantum mechanics (wave functions)
- Heat transfer and diffusion equations
- Circuit analysis and control systems
8.3 Computer Science
- Algorithm analysis and complexity
- Numerical computation and approximation
- Machine learning (optimization convergence)
9. References
- Rudin, W. (1976). Principles of Mathematical Analysis.
- Apostol, T. M. (1974). Mathematical Analysis.
- Bartle, R. G., & Sherbert, D. R. (2011). Introduction to Real Analysis.