Calculus II · Unit 4A · lesson
Sequences as Functions on the Integers
Treat a sequence as a function on integer inputs, read sequence notation, graph discrete terms, and distinguish the terms from a continuous guide curve.
Section overview
Sequences and their limitsWhat this section is building
Treat a sequence as a function on integer inputs, read sequence notation, graph discrete terms, and distinguish the terms from a continuous guide curve.
A sequence converges when every sufficiently late term remains arbitrarily close to one finite target.
Use algebraic limits when a formula is explicit; use bounds and monotonicity when a recurrence hides the formula.
Reading a finite plot as proof or solving a recurrence's fixed-point equation before proving convergence.
Learning objectives
interpret sequence notation, generate terms from a formula, and graph sequences as discrete points.
Sequences as Functions on the Integers
An infinite list still needs a precise rule
A sequence is an ordered list, but order alone is not enough for calculus. We treat a sequence as a function whose permitted inputs are positive integers. That viewpoint lets us use familiar function language - formula, domain, range, limit - while remembering that there are no sequence values between consecutive integers. A plotted sequence is therefore a collection of points rather than an unbroken curve.
The index carries meaning. In an application it may count payments, generations, bounces, or time steps. Changing the starting index can change notation without changing the underlying pattern, so every formula should be read together with its declared index set. The practical habit is simple: write several terms, label the starting index, and check that the formula actually matches the described list before attempting any limit.
From a list to a mathematical object
Start with the list . A pattern is visible, but calculus needs a rule that answers a sharper question: what is the term at position , and what happens when becomes arbitrarily large? Writing turns the pattern into a function rule and creates an object on which limit arguments can operate.
The discrete domain is not a minor technicality. We may draw a continuous guide curve to reveal shape, but the sequence has values only at its declared integer inputs. Throughout this unit, continuous functions will often help us reason about sequences, yet every conclusion must return to the actual integer-indexed terms.
Read this graph as text
The index selects one term at a time. A sequence is a function whose allowed inputs are integers. Each input selects one ordered term. Index-to-term machine plus discrete point graph. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in the index selects one term at a time; color is never the only cue.
Why it matters: Index-to-term machine plus discrete point graph.
A sequence is a function whose allowed inputs are integers. Each input selects one ordered term.
The index selects one term at a time. Index-to-term machine plus discrete point graph.
The order and the integer domain are part of the definition, not decoration.
Read this graph as text
A sequence is discrete, not a continuous curve. The points of a sequence occur only at positive integer inputs. A faint guide curve may reveal a pattern, but the curve is not part of the sequence. Index-to-term machine plus discrete point graph. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in a sequence is discrete, not a continuous curve; color is never the only cue.
Why it matters: Index-to-term machine plus discrete point graph.
The points of a sequence occur only at positive integer inputs. A faint guide curve may reveal a pattern, but the curve is not part of the sequence.
A sequence is discrete, not a continuous curve. Index-to-term machine plus discrete point graph.
The marked values rise toward the dashed level . The horizontal spacing is discrete: there is no sequence term at .
Core notation
A sequence may be written , , or as a rule such as
The index is an integer input; is the corresponding output.
Generate terms and identify a pattern
For , the first four terms are
The magnitudes rise toward , while the signs alternate. Writing terms before describing behavior prevents the common mistake of ignoring the factor .
One sequence can combine trend and oscillation
Consider . Its first terms are
The factor makes the terms alternate around , while makes the oscillation shrink. A complete description must mention both behaviors. Saying only that the sequence "goes up and down" misses the limiting structure.
A guide curve is not the sequence
Values such as do not exist for a sequence indexed by positive integers. A curve through the points may suggest behavior, but it does not add new terms.
u4a-sequences_as_functions-01For , find .
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Substitute into the rule.
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List the first five terms of .
Graph the first eight terms of .
Explain why a line drawn through sequence points is only a visual guide.
A bacteria count doubles each generation from an initial count of 50. Write a sequence formula.
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