Infinite Series Convergence & Power Series

1. Sequences and the Sequence of Partial Sums

An infinite sequence is an ordered map $a: \mathbb{N} \to \mathbb{R}$, denoted $(a_n)_{n=1}^\infty$. An infinite series is the formal sum $\sum_{n=1}^\infty a_n$.

Its convergence is defined strictly as the limit of its sequence of partial sums $s_N = \sum_{n=1}^N a_n$:

If $\lim_{N \to \infty} s_N$ does not exist as a finite real number, the series diverges.

2. The Necessary Divergence Test ($n$-th Term Test)

Contrapositive: If $\lim_{n \to \infty} a_n \ne 0$ (or does not exist), the series $\sum a_n$ diverges. Warning: The converse is false; the Harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ satisfies $\lim \frac{1}{n} = 0$ but diverges logarithmically.

3. The Cauchy Criterion for Series

This characterization relies solely on the completeness of $\mathbb{R}$, eliminating the need to know the sum $S$ a priori.

1. The Integral Test (Maclaurin-Cauchy)

Let $f: [1, \infty) \to \mathbb{R}$ be continuous, positive, and monotonically decreasing such that $f(n) = a_n$. Then:

Application to $p$-Series: $\sum_{n=1}^\infty \frac{1}{n^p}$ converges if and only if $p > 1$.

2. Direct Comparison & Limit Comparison Tests

• Direct: If $0 \le a_n \le b_n$, then $\sum b_n < \infty \implies \sum a_n < \infty$; and $\sum a_n = \infty \implies \sum b_n = \infty$.
• Limit: If $a_n > 0, b_n > 0$ and $L = \lim_{n \to \infty} \frac{a_n}{b_n} \in (0, \infty)$, then $\sum a_n$ and $\sum b_n$ share the exact same convergence status.

3. d'Alembert's Ratio Test

Let $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$:

• If $L < 1$: The series converges absolutely.
• If $L > 1$: The series diverges.
• If $L = 1$: The test is inconclusive (requires Raabe's test or Integral test).

4. Cauchy's Root Test

Let $L = \limsup_{n \to \infty} \sqrt[n]{|a_n|}$:

• If $L < 1$: Converges absolutely.
• If $L > 1$: Diverges.
• If $L = 1$: Inconclusive.

1. The Leibniz Alternating Series Test

An alternating series $\sum_{n=1}^\infty (-1)^{n-1} b_n$ with $b_n > 0$ converges if:

1. $b_{n+1} \le b_n$ for all $n \ge N$ (monotonic decrease).
2. $\lim_{n \to \infty} b_n = 0$.

Alternating Series Estimation Theorem: If $S$ is the sum, the truncation error $|R_N| = |S - s_N|$ satisfies:

The error is strictly bounded by the magnitude of the first neglected term.

2. Absolute vs Conditional Convergence

• Absolute Convergence: $\sum |a_n|$ converges. Every absolutely convergent series converges.
• Conditional Convergence: $\sum a_n$ converges, but $\sum |a_n| = \infty$ (e.g. the alternating harmonic series $\sum \frac{(-1)^{n-1}}{n} = \ln 2$).

3. Riemann's Rearrangement Theorem

If a series $\sum a_n$ is conditionally convergent, then for any real number $M \in \mathbb{R}$ (or $\pm\infty$), there exists a permutation $\sigma: \mathbb{N} \to \mathbb{N}$ of the series indices such that the rearranged series sums exactly to $M$:

In contrast, absolutely convergent series can be rearranged arbitrarily without altering their sum (Dirichlet's theorem).

1. Definition of a Power Series

A power series centered at $x_0 \in \mathbb{R}$ with real coefficients $(c_n)$ is:

2. The Cauchy-Hadamard Theorem

• The series converges absolutely for all $|x - x_0| < R$.
• The series diverges for all $|x - x_0| > R$.
• At the boundary points $x = x_0 \pm R$, the series may converge absolutely, converge conditionally, or diverge.

The Radius of Convergence $R$ is determined by the Cauchy-Hadamard formula:

The Interval of Convergence is one of $(x_0 - R, x_0 + R)$, $[x_0 - R, x_0 + R)$, $(x_0 - R, x_0 + R]$, or $[x_0 - R, x_0 + R]$.

1. Uniform Convergence on Compact Subsets

Within its open disk of convergence $|x - x_0| < R$, a power series converges uniformly on every compact subinterval $[x_0 - r, x_0 + r]$ ($r < R$). Consequently, $f(x)$ is infinitely differentiable ($C^\infty$) on $(x_0 - R, x_0 + R)$.

2. Term-by-Term Differentiation Theorem

The derivative of $f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n$ is obtained by differentiating term by term:

The derivative series has the exact same radius of convergence $R$.

3. Term-by-Term Integration Theorem

The indefinite integral of $f(x)$ is obtained by integrating term by term:

The integrated series also preserves the exact same radius of convergence $R$.