Series and Convergence: Finite and Infinite Series in Precalculus

Series and Convergence

Finite Series

Finite series are sums of a finite sequence of real numbers. They are foundational in algebra and calculus, providing a basis for understanding more complex infinite series.

• Definition: A finite series is written as $\sum_{i=m}^{n} a_i$, where $m, n \in \mathbb{Z}$ and $a_i$ are real numbers.

• Recursive Definition:

  • If $n < m$, $\sum_{i=m}^{n} a_i := 0$.

  • If $n \geq m-1$, $\sum_{i=m}^{n+1} a_i := \left(\sum_{i=m}^{n} a_i\right) + a_{n+1}$.

• Example: $\sum_{i=m}^{m+2} a_i = a_m + a_{m+1} + a_{m+2}$.

• Dummy Index: The index in the sum is a placeholder; $\sum_{i=m}^{n} a_i = \sum_{j=m}^{n} a_j$.

Basic Properties of Finite Sums

• Splitting an Interval: $\sum_{i=m}^{n} a_i + \sum_{i=n+1}^{p} a_i = \sum_{i=m}^{p} a_i$.

• Shift of Index: $\sum_{i=m}^{n} a_i = \sum_{j=m+k}^{n+k} a_{j-k}$ for any integer $k$.

• Linearity: $\sum_{i=m}^{n} (a_i + b_i) = \sum_{i=m}^{n} a_i + \sum_{i=m}^{n} b_i$.

• Scalar Multiplication: $\sum_{i=m}^{n} c a_i = c \sum_{i=m}^{n} a_i$ for $c \in \mathbb{R}$.

• Triangle Inequality: $\left|\sum_{i=m}^{n} a_i\right| \leq \sum_{i=m}^{n} |a_i|$.

• Monotonicity: If $a_i \leq b_i$ for all $i$, then $\sum_{i=m}^{n} a_i \leq \sum_{i=m}^{n} b_i$.

Summation over Finite Sets

Summation can be extended to finite sets using bijections, ensuring the sum is well-defined regardless of the order of elements.

• Definition: For a finite set $X$ with $n$ elements and $f: X \to \mathbb{R}$, choose a bijection $g: \{1, ..., n\} \to X$. Then $\sum_{x \in X} f(x) := \sum_{i=1}^{n} f(g(i))$.

• Example: If $X = \{a, b, c\}$ and $f(a) = 2, f(b) = 5, f(c) = -1$, then $\sum_{x \in X} f(x) = 2 + 5 - 1 = 6$.

• Properties:

  • If $X = \emptyset$, $\sum_{x \in X} f(x) = 0$.

  • If $X = \{x_0\}$, $\sum_{x \in X} f(x) = f(x_0)$.

  • Disjoint union: $\sum_{z \in X \cup Y} f(z) = \sum_{x \in X} f(x) + \sum_{y \in Y} f(y)$.

  • Linearity: $\sum_{x \in X} (f(x) + g(x)) = \sum_{x \in X} f(x) + \sum_{x \in X} g(x)$.

  • Monotonicity: If $f(x) \leq g(x)$ for all $x$, then $\sum_{x \in X} f(x) \leq \sum_{x \in X} g(x)$.

  • Triangle inequality: $|\sum_{x \in X} f(x)| \leq \sum_{x \in X} |f(x)|$.

Rearrangement of Finite Sums

Rearranging the terms of a finite sum does not affect its value, due to the properties of bijections and substitution.

• Rearrangement: For any permutation $f: X \to X$, $\sum_{i=n}^{m} a_i = \sum_{i=n}^{m} a_{f(i)}$.

Double Finite Series and Cartesian Products

Summing over Cartesian products allows for the evaluation of double series, which is foundational for understanding multidimensional sums.

• Lemma: $\sum_{x \in X} \left(\sum_{y \in Y} f(x, y)\right) = \sum_{(x, y) \in X \times Y} f(x, y)$.

• Fubini's Theorem (Finite Sums): Changing the order of summation is always valid for finite sums.

Infinite Series

Formal Infinite Series

An infinite series is an expression of the form $\sum_{n=m}^{\infty} a_n$, where $a_n$ are real numbers. The concept of convergence is central to determining whether such a series represents a meaningful sum.

• Partial Sums: $S_N := \sum_{n=m}^{N} a_n$.

• Convergence: The series converges to $L$ if $\lim_{N \to \infty} S_N = L$.

• Divergence: If the partial sums do not approach a finite limit, the series is divergent.

• Uniqueness: The sum of a convergent series is unique.

Examples

• Convergent Geometric Series: $\sum_{n=1}^{\infty} 2^{-n}$ converges to $1$.

• Divergent Geometric Series: $\sum_{n=1}^{\infty} 2^n$ diverges.

• p-Series: $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, but $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.

Cauchy-Type Criterion for Series

A series converges if and only if its tails can be made arbitrarily small, formalized by the Cauchy criterion.

• Criterion: For every $\varepsilon > 0$, there exists $N$ such that $\left|\sum_{n=p}^{q} a_n\right| \leq \varepsilon$ for all $p, q \geq N$.

Zero Test

If a series converges, its terms must approach zero.

• Test: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum_{n=m}^{\infty} a_n$ diverges.

• Example: $\sum_{n=1}^{\infty} 1$ diverges; $\sum_{n=1}^{\infty} (-1)^n$ diverges.

• Warning: The converse is not true; $\lim a_n = 0$ does not guarantee convergence (e.g., harmonic series).

Absolute and Conditional Convergence

Absolute convergence is a stronger form of convergence, ensuring the sum of absolute values also converges.

• Absolute Convergence: $\sum_{n=m}^{\infty} |a_n|$ converges.

• Test: If $\sum |a_n|$ converges, then $\sum a_n$ converges and $\left|\sum a_n\right| \leq \sum |a_n|$.

• Conditional Convergence: $\sum a_n$ converges, but $\sum |a_n|$ does not.

• Example: Alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ is conditionally convergent.

Alternating Series Test

Alternating series with decreasing, non-negative terms converging to zero are convergent.

• Test: If $a_n \geq 0$, $a_n \geq a_{n+1}$, and $\lim_{n \to \infty} a_n = 0$, then $\sum_{n=m}^{\infty} (-1)^n a_n$ converges.

• Example: Alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges.

Series Laws

• Linearity: $\sum (a_n + b_n) = \sum a_n + \sum b_n$; $\sum c a_n = c \sum a_n$.

• Index Shift: Convergence depends only on the tail of the series; shifting indices does not affect convergence.

• Telescoping Series: $\sum_{n=0}^{\infty} (a_n - a_{n+1})$ converges to $a_0$ if $a_n \to 0$.

Series with Non-Negative Terms

Monotone Partial Sums

For series with non-negative terms, partial sums are increasing. Convergence occurs if the sequence of partial sums is bounded above.

• Characterization: $\sum_{n=m}^{\infty} a_n$ converges if and only if $\sum_{n=m}^{N} a_n \leq M$ for all $N \geq m$.

Comparison Test

If the terms of one series are dominated by those of a convergent series, then the dominated series is also absolutely convergent.

• Test: If $|a_n| \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ is absolutely convergent.

Geometric Series

• Test: $\sum_{n=0}^{\infty} x^n$ converges if $|x| < 1$ and $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$.

Cauchy Criterion for Decreasing Non-Negative Terms

For decreasing non-negative sequences, convergence can be tested using dyadic indices.

• Test: $\sum_{n=1}^{\infty} a_n$ converges if and only if $\sum_{k=0}^{\infty} 2^k a_{2^k}$ converges.

p-Series Test

• Test: $\sum_{n=1}^{\infty} \frac{1}{n^q}$ converges if $q > 1$, diverges if $q \leq 1$.

• Riemann Zeta Function: $\zeta(q) = \sum_{n=1}^{\infty} \frac{1}{n^q}$, central in number theory.

Rearrangement of Series

Finite vs Infinite Sums

• Finite Sums: Order does not matter.

• Infinite Series: Rearrangement can affect convergence and sum, especially for conditionally convergent series.

Rearrangement of Non-Negative and Absolutely Convergent Series

• Non-Negative Series: Rearrangement preserves convergence and sum.

• Absolutely Convergent Series: Rearrangement preserves convergence and sum.

Conditionally Convergent Series

• Rearrangement: Can change the sum or even cause divergence.

• Riemann's Theorem: For a conditionally convergent series, rearrangement can yield any real sum or divergence.

The Root and Ratio Tests

Root Test

The root test determines convergence based on the nth root of the absolute value of terms.

• Test: Let $\alpha = \limsup_{n \to \infty} |a_n|^{1/n}$.

  • If $\alpha < 1$, the series is absolutely convergent.

  • If $\alpha > 1$, the series is divergent.

  • If $\alpha = 1$, the test is inconclusive.

Ratio Test

The ratio test uses the ratio of consecutive terms to determine convergence.

• Test:

  • If $\limsup_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} < 1$, the series is absolutely convergent.

  • If $\liminf_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} > 1$, the series is divergent.

  • Otherwise, the test is inconclusive.

Examples and Applications

• Example: $\sum_{n=1}^{\infty} n^q x^n$ with $|x| < 1$ converges absolutely by the ratio test.

• Example: $\lim_{n \to \infty} n^{1/n} = 1$.

Other Convergence Tests

Root and ratio tests are most effective for series with geometric-like terms. For other series, such as those involving $1/n^q$, the integral test and other methods may be more suitable.

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