Difficulty: Foundational. Prerequisites: MATH 103 or equivalent Calculus 1 (derivatives, basic integration, function behaviour).
This is where Calculus 2 begins. Limits underpin every major concept in calculus, from derivatives to integrals, and sequences and series extend those ideas to infinite processes. If you are coming in cold, the core question is: what happens to a function or a list of numbers as you keep going, closer and closer to some target, or further and further out? That question, made precise, is the concept of a limit. Sequences and series then ask what happens when you line up infinitely many values, or add them all together. These ideas are the backbone of Taylor series, convergence tests, and power series, all of which dominate the rest of MATH 104.
A limit describes the value a function or sequence settles toward as its input moves closer to some target. A sequence is an ordered list of numbers (possibly infinite), and a series is what you get when you add up the terms of a sequence. These three ideas are the foundation of Calculus 2: every convergence test, Taylor expansion, and power series you will meet this semester depends on them.
Limit (Convergence)
The value that a function or sequence approaches as the input or index approaches some target value. Limits are used to define continuity, derivatives, and integrals, and they can resolve classic paradoxes such as Zeno's paradoxes of motion.
In simple terms, a limit answers the question: "What value is this thing heading toward, even if it never quite arrives?"
Sequence
An ordered collection of objects (called elements, members, or terms) in which repetitions are allowed. The number of elements, which may be infinite, is called the sequence's length. Order matters, and the same element can appear at different positions.
Think of it as a numbered list that can go on forever: a₁, a₂, a₃, ... Each slot has exactly one value, but the same value can show up in more than one slot.
Series
The sum of the terms of a sequence. If the sequence is {a₁, a₂, a₃, ...}, the corresponding series is a₁ + a₂ + a₃ + ...
In simple terms, a series is what happens when you start adding up every entry in a sequence, one after another. The central question is whether that running total settles on a finite number (converges) or grows without bound (diverges).
Convergence
A sequence or series converges if it approaches a specific finite value as the number of terms increases. If it does not, it diverges.
Think of it as: does the sum (or the list) eventually calm down and stay near one number, or does it wander off or blow up?
Divergence
A sequence or series diverges if it does not approach a specific finite value. This can mean it grows to infinity, oscillates, or otherwise fails to settle.
Infinitesimal
A quantity that is closer to zero than any positive real number but is not itself zero. Newton and Leibniz both used infinitesimals as the conceptual building blocks of calculus, though the rigorous formalisation came later with limits.
In simple terms, an infinitesimal is an "infinitely small" quantity, too small to measure but not nothing.
A limit captures what value a function f(x) or a sequence aₙ is heading toward, without requiring that it ever reaches that value.
Limits are used to define three pillars of calculus: continuity (a function is continuous at a point if the limit equals the function value), derivatives (defined as the limit of a difference quotient), and integrals (defined as the limit of Riemann sums).
A limit can exist even when the function is undefined at the target point. For example, (sin x)/x is undefined at x = 0, but its limit as x approaches 0 is 1.
Limits also resolve Zeno's paradoxes: the infinite sum 1/2 + 1/4 + 1/8 + ... converges to 1, so traversing "infinitely many" sub-distances still covers a finite distance.
A sequence is written {a₁, a₂, a₃, ...} or {aₙ} for short.
Order matters. The sequence {1, 2, 3} is different from {3, 2, 1}.
Repetitions are allowed. {1, 1, 1, ...} is a perfectly valid (constant) sequence.
A sequence converges if its terms approach a single finite value L as n grows: for any small tolerance ε > 0, eventually every term stays within ε of L.
A sequence that does not converge diverges. It might tend to infinity, oscillate (like (-1)ⁿ), or behave chaotically.
A series is the sum of a sequence's terms: S = Σ aₙ from n = 1 to ∞.
The partial sums S₁ = a₁, S₂ = a₁ + a₂, S₃ = a₁ + a₂ + a₃, ... form their own sequence.
A series converges if and only if the sequence of partial sums converges to a finite value.
The geometric series Σ rⁿ converges to 1/(1 - r) when |r| < 1 and diverges otherwise. This is the simplest and most important convergent series in the course.
The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges, even though its terms approach zero. This is a critical example: terms going to zero is necessary but not sufficient for convergence.
Isaac Newton (1642 - 1727): English physicist and mathematician, one of two independent inventors of modern calculus in the 17th century. He applied calculus to his laws of motion and gravitation, introduced the product rule, chain rule, higher derivatives, Taylor series, and analytical functions. His notation (dot notation for derivatives) was idiosyncratic. Major work: Principia Mathematica (1687). He also developed series expansions for functions with fractional and irrational powers.
Gottfried Leibniz (1646 - 1716): German polymath and philosopher, the other independent inventor of calculus. He was the first to publish his results and is now recognised as an independent inventor and contributor. His key contribution was a clear, systematic set of rules for working with infinitesimal quantities, including the product rule and chain rule in differential and integral forms. He developed much of the notation used in calculus today (dy/dx, the integral sign ∫). He spent considerable effort on choosing good symbols, and his notation won out over Newton's for most purposes.
Both Newton and Leibniz used infinitesimals, which were considered conceptually suspect at the time. The rigorous ε-δ definition of limits, which replaced infinitesimals, came later with Cauchy and Weierstrass in the 19th century.
Limit notation
lim(x→a) f(x) = L means f(x) approaches L as x approaches a.
lim(n→∞) aₙ = L means the sequence {aₙ} converges to L.
Formal (ε-δ) definition of a limit
For every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
For sequences: for every ε > 0, there exists a positive integer N such that if n > N, then |aₙ - L| < ε.
Series (sigma notation)
Σ (n=1 to ∞) aₙ = a₁ + a₂ + a₃ + ...
The partial sum: Sₙ = Σ (k=1 to n) aₖ
Geometric series formula
Σ (n=0 to ∞) rⁿ = 1/(1 - r), valid when |r| < 1
More generally: Σ (n=0 to ∞) arⁿ = a/(1 - r), valid when |r| < 1
Leibniz notation for derivatives
dy/dx (first derivative), d²y/dx² (second derivative). The integral sign ∫ was introduced by Leibniz as an elongated S, standing for "summa."
Limits are the mathematical machinery behind any process that involves approaching a value without necessarily reaching it. Engineers use limits when modelling material stress at a breaking point, and economists use them to describe marginal cost and revenue as production scales.
Series appear whenever you approximate a complicated function with simpler pieces. Your calculator computes sin(x) and eˣ using truncated Taylor series. Signal processing (music streaming, image compression) relies on Fourier series, which decompose complex waveforms into sums of sine and cosine terms. Compound interest, computed continuously, is derived from the limit definition of the exponential function.
Students often think that if the terms of a series approach zero, the series must converge. It does not. The harmonic series (1 + 1/2 + 1/3 + ...) is the classic counterexample: every term shrinks toward zero, but the sum diverges. The nth-term test only tells you a series diverges when terms do not go to zero; it cannot confirm convergence.
Students sometimes confuse sequences and series. A sequence is the list; a series is the sum. The sequence {1, 1/2, 1/4, 1/8, ...} converges to 0. The series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. Different objects, different limits.
Students often believe a limit must be "reached" by the function. A function can have a limit at a point where it is undefined or where it takes a different value. The limit is about the trend, not the arrival.
Students sometimes assume Newton invented calculus alone. Newton and Leibniz developed it independently. Leibniz published first, and his notation (dy/dx, ∫) is what we use today. The priority dispute between them is one of the most famous controversies in the history of mathematics.
⚠️ The distinction between a sequence converging and a series converging is a favourite exam question. Know both definitions cold.
⚠️ The geometric series formula (a/(1 - r) when |r| < 1) is tested repeatedly. Be able to identify the first term a and the common ratio r from a series that is not already in standard form.
⚠️ The harmonic series diverging despite terms going to zero is a standard "explain why" question. Be ready to state that the nth-term test is necessary but not sufficient.
⚠️ Expect at least one problem asking you to compute a limit using the formal ε-δ definition or the sequence definition (given ε, find N). Practise the algebra of solving for δ or N.
⚠️ Partial sums are the bridge between sequences and series. If an exam asks "does this series converge?", you are being asked whether the sequence of partial sums has a finite limit.
True or false: If the terms of a series approach zero, the series converges. (Answer: False. The harmonic series is the counterexample.)
Fill in the blank: A series converges if and only if the sequence of ______ converges. (Answer: partial sums.)
True or false: The geometric series Σ rⁿ converges for r = 1. (Answer: False. It converges only when |r| < 1.)
Fill in the blank: ______ developed much of the notation used in calculus today, including dy/dx and the integral sign. (Answer: Gottfried Leibniz.)
True or false: A function must be defined at a point for the limit to exist at that point. (Answer: False. The limit depends on the behaviour near the point, not at it.)
Q: State the formal definition of the limit of a sequence. What does it mean for {aₙ} to converge to L?
A: For every ε > 0, there exists a positive integer N such that for all n > N, |aₙ - L| < ε. In plain language, no matter how tight a band you draw around L, eventually all terms of the sequence stay inside that band.
Q: Does the series Σ (n=1 to ∞) 1/n converge or diverge? Justify your answer.
A: It diverges. This is the harmonic series. Even though the terms 1/n approach zero, the partial sums grow without bound. The nth-term test does not help here (it only confirms divergence when terms do not approach zero), but comparison with the integral of 1/x, or grouping terms, shows divergence.
Q: Compute the sum of the geometric series 3 + 3/2 + 3/4 + 3/8 + ...
A: This is a geometric series with first term a = 3 and common ratio r = 1/2. Since |r| < 1, it converges to a/(1 - r) = 3/(1 - 1/2) = 3/(1/2) = 6.
Q: Explain the difference between a sequence and a series.
A: A sequence is an ordered list of terms: a₁, a₂, a₃, ... A series is the sum of those terms: a₁ + a₂ + a₃ + ... A sequence can converge (its terms approach a value) while the corresponding series diverges (their sum grows without bound), as with the sequence {1/n} and the harmonic series.
Q: Why is Leibniz's notation (dy/dx) used more widely than Newton's (dot notation)?
A: Leibniz designed his notation to make the chain rule and substitution visually intuitive. Writing dy/dx suggests a ratio of infinitesimal changes, which naturally extends to expressions like (dy/dx)(dx/dt) = dy/dt. Newton's dot notation (ẏ) is compact but does not generalise as cleanly to partial derivatives, higher dimensions, or the chain rule, so Leibniz's notation became the standard for most of calculus.
Limits, sequences, and series feed directly into the convergence tests you will study next (ratio test, root test, comparison test, integral test), all of which ask whether a given series converges or diverges. Taylor and Maclaurin series, which represent functions as infinite polynomials, depend entirely on understanding when a power series converges. Improper integrals, another major Calculus 2 topic, are defined as limits and often require the same convergence reasoning. In linear algebra and differential equations, series solutions (Frobenius method, for instance) build on the same machinery.
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