Calculus: Sequences and Series

Sequences and series form the bridge between finite algebra and the infinite processes at the heart of calculus, enabling us to rigorously analyze and sum an endless list of numbers. Mastering this area is essential for approximating complex functions, solving otherwise intractable differential equations, and modeling phenomena in engineering and physics where simple formulas fall short.

From Sequences to Series: The Foundation of Infinite Sums

A sequence is an ordered list of numbers, $a_1,a_2,a_3,...$, often defined by an explicit formula for the $n$th term, $a_n$. The central question for a sequence is whether it converges—does it approach a specific finite limit $L$ as $n$ tends to infinity? Formally, a sequence $\{a_n\}$ converges to $L$ if for every $ϵ>0$, there exists an integer $N$ such that for all $n>N$, $|a_n-L|<ϵ$. If no such limit exists, the sequence diverges.

A series is the sum of the terms of a sequence. Given a sequence $\{a_n\}$, we form the associated series $a_1+a_2+a_3+...$, denoted ${\sum }_{n=1}^{\infty }{a}_n$. Its convergence is defined via the sequence of partial sums, $S_N={\sum }_{n=1}^N{a}_n$. The series converges to a sum $S$ if the sequence of partial sums $\{S_N\}$ converges to $S$. A fundamental starting point is the geometric series ${\sum }_{n=0}^{\infty }a{r}^n$, which converges to $\frac{a}{1-r}$ if $|r|<1$ and diverges otherwise. Another crucial, counterintuitive result is the harmonic series ${\sum }_{n=1}^{\infty }\frac{1}{n}$, which famously diverges despite its terms approaching zero.

The Toolkit: Convergence Tests for Series

Determining convergence directly from the partial sum definition is often impossible. Instead, we use a battery of tests, each with specific strengths for different series types.

The Integral Test: Suppose $f(x)$ is a continuous, positive, decreasing function for $x\ge 1$ and $a_n=f(n)$. Then the series ${\sum }_{n=1}^{\infty }{a}_n$ converges if and only if the improper integral ${\int }_1^{\infty }f(x)dx$ converges. This test is powerful for series like the p-series $\sum \frac{1}{n^p}$, which converges if $p>1$ and diverges if $p\le 1$, a result proven using this test.

The Comparison Tests: These are workhorses for series with non-negative terms. The Direct Comparison Test states that if $0\le a_n\le b_n$ for all $n$, and $\sum b_n$ converges, then $\sum a_n$ converges. Conversely, if $\sum a_n$ diverges, then $\sum b_n$ diverges. The Limit Comparison Test is often more practical: if ${\lim }_{n\to \infty }\frac{a_n}{b_n}=L$, where $0<L<\infty$, then both series $\sum a_n$ and $\sum b_n$ converge or diverge together. You might compare a complicated rational expression to a simpler p-series.

The Ratio and Root Tests: These are ideal for series involving factorials, exponentials, or $n$th powers.

• Ratio Test: For a series $\sum a_n$, compute ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=L$. If $L<1$, the series converges absolutely; if $L>1$, it diverges; if $L=1$, the test is inconclusive. This test is particularly effective for series like $\sum \frac{n!}{10^n}$.
• Root Test: Compute ${\lim }_{n\to \infty }\sqrt[n]{|a_n|}=L$. The same convergence/divergence rules as the Ratio Test apply. It is often used when each term $a_n$ involves an expression raised to the $n$th power, such as in $\sum {\left(\frac{3n+2}{5n+1}\right)}^n$.

Power Series, Taylor Series, and Function Representations

A power series is an infinite series of the form ${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n$, centered at $x=a$. Its behavior is characterized by its radius of convergence, $R$ ($0\le R\le \infty$). For $|x-a|<R$, the series converges absolutely; for $|x-a|>R$, it diverges. You find $R$ using the Ratio or Root Test (applied to the coefficients $c_n$). The interval of convergence is the set of all $x$ for which the series converges, which you must check separately at the endpoints $x=a\pm R$.

When a power series represents a function, it provides a powerful tool for approximation. The Taylor series of a function $f(x)$ infinitely differentiable at $a$ is its representation as a specific power series: $$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n.$$ When the center is $a=0$, the series is called a Maclaurin series. For example, the Maclaurin series for $e^x$ is ${\sum }_{n=0}^{\infty }\frac{x^n}{n!}$, which has an infinite radius of convergence. The Taylor polynomial $T_N(x)$ is the $N$th partial sum of this series, providing a polynomial approximation to $f(x)$. The error of this approximation is given by the remainder term, $R_N(x)=f(x)-T_N(x)$, often bounded using Taylor's Inequality.

Applications: From Approximation to Differential Equations

The utility of series extends far beyond abstract convergence questions. Series expansions allow us to approximate non-polynomial functions with polynomials, enabling calculations in physics and engineering. For instance, for small $x$, $\sin x\approx x-\frac{x^3}{6}$, an approximation derived from its Maclaurin series.

One of the most powerful applications is solving differential equations. The power series method involves assuming a solution to a differential equation can be written as $y={\sum }_{n=0}^{\infty }{c}_n{x}^n$. You substitute this series into the equation, combine like powers of $x$, and derive a recurrence relation for the coefficients $c_n$. This method is indispensable for equations with variable coefficients (like Bessel's equation) that lack solutions in terms of elementary functions.

Common Pitfalls

1. Misapplying the $n$th Term Test for Divergence: Remember, if ${\lim }_{n\to \infty }{a}_n=0$, the series $\sum a_n$ may converge (e.g., a p-series with $p>1$) or diverge (e.g., the harmonic series). The test only states that if the limit is not zero, the series must diverge. Using it as a test for convergence is a critical error.
2. Confusing Sequence and Series Convergence: A series $\sum a_n$ can only converge if its underlying sequence of terms $\{a_n\}$ converges to zero. However, the convergence of the sequence $\{a_n\}$ to zero does not guarantee the convergence of the series. The harmonic series is the classic counterexample.
3. Incorrectly Handling the Comparison Tests: For the Direct Comparison Test, you must establish the inequality $0\le a_n\le b_n$ before drawing a conclusion. You cannot compare a divergent series to a larger divergent series to prove divergence of the smaller one. For the Limit Comparison Test, the limit $L$ must be a finite, positive number ($0<L<\infty$). If $L=0$ or $L=\infty$, the test is inconclusive.
4. Forgetting to Check Endpoints for Interval of Convergence: After using the Ratio or Root Test to find the radius of convergence $R$, you must test the series at the endpoints $x=a\pm R$ separately. A power series may converge at one, both, or neither endpoint. The interval of convergence is not determined by the radius alone.

Summary

• A series is the sum of a sequence. It converges if its sequence of partial sums converges to a finite limit.
• Convergence tests—including the Integral, Comparison (Direct and Limit), Ratio, and Root tests—provide a systematic toolkit for analyzing series with non-negative or absolute terms.
• A power series $\sum c_n(x-a{)}^n$ defines a function within its interval of convergence, the center of which is determined by its radius of convergence, $R$.
• The Taylor series and Maclaurin series (centered at 0) provide polynomial representations of functions, with the coefficients derived from the function's derivatives at the center.
• Series are applied to approximate functions and to find solutions to differential equations through the power series method, offering solutions where traditional algebraic techniques fail.