What Are Series & Sequences — and Why Do They Matter?
An infinite sum that actually converges to a finite number is one of the most beautiful ideas in mathematics. This section builds that idea from scratch.
A sequence is simply an ordered list of numbers: a₁, a₂, a₃, … The central question is whether the terms approach a limit as the index grows: and if so, what that limit is. A series takes this further: it adds the terms of a sequence together and asks whether the running total settles on a finite value.
This might seem abstract, but the applications are everywhere. The number e is defined as the sum 1 + 1 + 1/2! + 1/3! + …, an infinite series. Your calculator evaluates sin(x) by summing a few terms of the sine series and stopping when the error is small enough. Signal processing, quantum mechanics, financial mathematics, and image compression all rest on series expansions of functions.
At varsity level, two big skills dominate this section. First, you need to decide whether a given series converges (adds to a finite number) or diverges (blows up or oscillates forever). There are six main tests for this, each suited to a different family of series. Second, you need to represent functions as infinite polynomials, power series, and in particular Taylor and Maclaurin series, and then use them to approximate values, evaluate tricky limits, and integrate functions that have no elementary antiderivative.
All 10 Topics in This Section
Work through them in order if you are new to the subject. Each page stands alone if you need a specific topic fast.
Convergence Tests — Quick Reference
Six tests, each with a different sweet spot. The skill is knowing which to reach for first.
No single convergence test works on every series. Part of mastering this section is building intuition for which test fits which series: a rational-looking series calls for comparison, a series with factorials calls for the ratio test, and a series that alternates sign calls for Leibniz's test.
| Test | Condition for Convergence | Best Used When |
|---|---|---|
| Divergence Test | If lim aₙ ≠ 0, series diverges. (Cannot prove convergence.) | Always try this first — it is the cheapest check. |
| Geometric Series | Σ arⁿ converges iff |r| < 1. Sum = a/(1−r). | Terms are a constant raised to a power of n. |
| p-Series Test | Σ 1/nᵖ converges iff p > 1. | Terms look like 1/nᵖ or 1/n^(something). |
| Direct Comparison | If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, so does Σaₙ. | Terms can be bounded above/below by a known series. |
| Limit Comparison | If lim aₙ/bₙ = c > 0, both series behave the same. | Rational-type terms where direct comparison is awkward. |
| Ratio Test | L = lim|aₙ₊₁/aₙ|. Converges if L < 1, diverges if L > 1. | Terms contain n! or aⁿ (exponentials and factorials). |
| Root Test | L = lim|aₙ|^(1/n). Same conclusion rule as ratio test. | Terms are an nth power: (f(n))ⁿ. |
| Alternating Series Test | If terms decrease to 0 in magnitude, the alternating series converges. | Series alternates sign: (−1)ⁿ aₙ with aₙ → 0 decreasing. |
Power Series and Taylor Expansions
The second half of this section moves beyond convergence testing into representing functions as infinite polynomials, one of the most powerful tools in all of analysis.
A power series centred at a is a series of the form Σcₙ(x − a)ⁿ, where the cₙ are constants. Unlike the series in §03, a power series defines a function of x, it converges on some interval (a − R, a + R), where R is the radius of convergence. Inside that interval, you can differentiate or integrate the series term by term, just like a polynomial.
The most important power series are Taylor series, where the coefficients are determined by the derivatives of a known function:
Coefficient of (x−a)ⁿ = f⁽ⁿ⁾(a) ÷ n!
When the centre is a = 0, this becomes a Maclaurin series. The five series below appear in almost every problem, you should know them from memory:
sin x = x − x³/3! + x⁵/5! − … (R = ∞)
cos x = 1 − x²/2! + x⁴/4! − … (R = ∞)
ln(1+x) = x − x²/2 + x³/3 − … (R = 1)
1/(1−x) = 1 + x + x² + x³ + … (R = 1)
Once you know these five, you can derive almost any other Maclaurin series by substitution, multiplication, differentiation, or integration, without ever computing a single derivative from scratch.
Using Taylor Series to Evaluate Limits
Taylor series give an elegant way to resolve indeterminate limits that would require many applications of L'Hôpital's rule. The strategy is to expand numerator and denominator as series near the limit point, cancel the leading terms, and read off the answer.
For example, lim_{x→0} (sin x − x) / x³: expanding sin x gives (x − x³/6 + …) − x = −x³/6 + … Dividing by x³ yields −1/6 as x → 0. Three rounds of L'Hôpital would give the same result far more tediously.
Recommended Learning Path
If you are working through this section for the first time, follow this order. Each step builds directly on the previous one.
Understand what it means for a sequence to converge before worrying about infinite sums.
Learn the definition of a convergent series via partial sums, and the divergence test as a first filter.
The one series family you can sum exactly. Essential for comparison later.
The second benchmark series: Σ1/nᵖ. Pairs with the geometric series as your main comparison targets.
Now that you know what to compare against, these tests handle most rational-looking series.
Go-to tools for series involving factorials, exponentials, or nth powers.
Handle sign-alternating series and understand why absolute convergence is the gold standard.
Bridge from numerical series to functions of x. Learn how to find and use the interval of convergence.
The crown jewel. Derive series from derivatives, bound the error, use series to evaluate limits and non-elementary integrals.
Master the a = 0 special case, build a reference table, and learn to derive new series without computing derivatives.
Frequently Asked Questions
How This Section Connects to the Rest of Calculus
Series and sequences do not exist in isolation, they are tightly woven through the rest of the curriculum.
Limits are foundational: the definition of sequence convergence and partial-sum convergence both rely on the epsilon-delta limit concept. L'Hôpital's rule and Taylor series complement each other for evaluating indeterminate limits: knowing when to use each is a key exam skill.
Differentiation feeds directly into Taylor series via repeated derivatives. Higher-order derivatives are the machinery that produces Taylor coefficients. Logarithmic and implicit differentiation come up when deriving less obvious series.
Integration connects through the fundamental theorem: the integral test for convergence, and the technique of integrating a power series term by term to evaluate integrals like ∫ sin(x²) dx or ∫ e^(−x²) dx that cannot be expressed in elementary closed form. See improper integrals for the integral test background.
Differential equations use power series as a method of solution: when other techniques fail, you assume a power series solution Σcₙxⁿ and solve for the coefficients. This is how Bessel functions and Legendre polynomials are defined.