Convergence of Sequences and Series

In real analysis, we move beyond calculating limits to proving their existence and understanding the properties of the spaces they inhabit. This lesson focuses on the behavior of infinite sequences and the criteria for the summation of infinite series.

Sequence Convergence

A sequence $(a_n)$ in $\mathbb{R}$ converges to $L$ if for every $ϵ>0$, there exists an $N\in \mathbb{N}$ such that for all $n>N$: $|a_n-L|<ϵ$

Important Theorems for Sequences

1. Monotone Convergence Theorem: Every bounded monotonic sequence in $\mathbb{R}$ is convergent. This is a direct consequence of the Completeness Axiom (the supremum property).
2. Bolzano-Weierstrass Theorem: Every bounded sequence in $\mathbb{R}$ has a convergent subsequence. This is a foundational result for compactness.
3. Cauchy Sequences: A sequence is Cauchy if its terms become arbitrarily close to each other: $\forall ϵ>0\exists N:n,m>N⟹|a_n-a_m|<ϵ$.
   • Completeness: In $\mathbb{R}$, a sequence converges if and only if it is a Cauchy sequence.

Infinite Series

A series $\sum a_n$ is the limit of its partial sums $S_k={\sum }_{n=1}^k{a}_n$. If $\lim S_k=S$, the series converges to $S$.

Convergence Tests

To determine if a series converges without finding its sum, we use several tests:

• Comparison Test: If $0\le a_n\le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.
• Ratio Test: $L={\lim }_{n\to \infty }|a_{n+1}/a_n|$. If $L<1$ it converges; if $L>1$ it diverges.
• Root Test: $L={\lim }_{n\to \infty }\sqrt[n]{|a_n|}$. If $L<1$ it converges.
• Integral Test: If $f(x)$ is positive, continuous, and decreasing, then $\sum f(n)$ converges if and only if ${\int }_1^{\infty }f(x)dx$ converges.

Absolute vs. Conditional Convergence

• Absolute Convergence: $\sum a_n$ converges absolutely if $\sum |a_n|$ converges. Absolute convergence implies convergence.
• Conditional Convergence: $\sum a_n$ converges, but $\sum |a_n|$ diverges (e.g., the Alternating Harmonic Series $\sum \frac{(-1{)}^{n+1}}{n}$).

Rearrangements

A shocking result in analysis is Riemann’s Rearrangement Theorem: If a series is conditionally convergent, its terms can be rearranged to sum to any real value, or even to diverge. In contrast, absolutely convergent series always sum to the same value regardless of order.

Taylor Series

For a smooth function $f(x)$, the Taylor series at $a$ is: $f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$ A function is analytic if it is represented by its Taylor series in some neighborhood. Not all smooth functions are analytic!

Python: Testing Numerical Convergence

We can use Python to estimate the limit of a series and observe the rate of convergence for the Ratio Test.

import math

def series_sum(func, n_terms):
    total = 0
    for n in range(1, n_terms + 1):
        total += func(n)
    return total

# Harmonic series (divergent)
harmonic = lambda n: 1/n
# Geometric series (convergent) 1/2^n
geometric = lambda n: 1/(2**n)

print(f"Partial sum of harmonic (1000 terms): {series_sum(harmonic, 1000)}")
print(f"Partial sum of geometric (1000 terms): {series_sum(geometric, 1000)}")
# Note how the geometric series quickly approaches 1.0

Significance

Sequences and series allow us to define transcendental functions like $\sin (x)$, $e^x$, and $\ln (x)$ using only basic arithmetic operations. They are the language of approximation and numerical methods, forming the bridge between discrete logic and continuous reality.

Exercise