Defining Convergent and Divergent Infinite Series

Distinguish a term sequence from partial sums and use their finite limit to define a series sum.

College Board Topic 10.1 · CED p. 181 (PDF p. 186) · CED effective Fall 2020; project course outline 2026–27.

A sequence is an ordered list $a_{1}, a_{2}, \dots$ . A series adds that list. Since an infinite addition is not a finished arithmetic operation, define the nth partial sum $S_{n} = {\sum_{k = 1}}^{n} a_{k}$ . The infinite series converges to S exactly when the sequence of partial sums has a finite real limit S. If that limit is infinite or does not exist, the series diverges.

Keep two limits separate: the limit of the terms aₙ and the limit of the running totals Sₙ. Terms becoming tiny is necessary for a convergent series, but does not tell us whether their accumulated total stays finite. Watch a graph of terms and a graph of partial sums side by side; they answer different questions.

For $a_{n} = 1 / (n (n + 1))$ , split the term as $1 / n - 1 / (n + 1)$ . In the finite partial sum, the middle terms cancel, leaving $S_{n} = 1 - 1 / (n + 1)$ . The limit of these running totals is 1. The cancellation is justified in a finite sum before we take a limit, so we have actually proved the sum rather than trusting a pattern of rounded decimals.

Read more in the retained sequence lesson and infinite-series lesson.

Terms and running totals for 1/(n(n+1))
n	aₙ	Sₙ
1	1/2	1/2
2	1/6	2/3
3	1/12	3/4
4	1/20	4/5

Let's work through it

Write $S_{n} = {\sum_{k = 1}}^{n} (1 / k - 1 / (k + 1))$ . Why: A partial sum contains finitely many terms.
Cancel to get $S_{n} = 1 - 1 / (n + 1)$ . Why: Every interior reciprocal appears once positively and once negatively.
Take the limit: $S_{n} \to 1$ . Why: A finite limit of partial sums defines the series sum.

Try it from memory

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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.

Question 1 of 20 self-assessed

Question 1

S_{n} = 5 - 1 / n

, what is the infinite sum?

Your answer and reasoning

JavaScript enables the written-work controls. You can still try the questions below on paper and then open their reasoning.

Question 1: prompt and worked reasoning

S_{n} = 5 - 1 / n

, what is the infinite sum?

5

The partial sums have finite limit 5.

Question 2: prompt and worked reasoning

If aₙ tends to zero, have you shown that the series converges?

No.

The series is governed by the limit of Sₙ, not solely the term limit.

See every recall question with its answer

These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.

Retrieval 1

S_{n} = 5 - 1 / n

, what is the infinite sum?

Worked answer and reason

5

The partial sums have finite limit 5.

Retrieval 2

If aₙ tends to zero, have you shown that the series converges?

Worked answer and reason

No.

The series is governed by the limit of Sₙ, not solely the term limit.

Catch the tempting mistake

A series converges when its terms converge.: The running totals must converge to a finite number. A convergent term sequence can still accumulate an infinite total.

Make the picture move

BC: Taylor series, convergence, and error

A polynomial is local evidence, not a universal replacement.

Try first. Predict how the approximation changes near zero and near x = 3. Compare actual error with the guaranteed bound.

sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} x^{2 k + 1}}{(2 k + 1)!}

Choose a scenario Nonzero terms m 3 Evaluate at x 0.75

Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.

Function: sin x
Taylor polynomial: degree 5

The solid function curve is compared with a dashed Taylor polynomial. Filled and open markers compare the selected values. At x = 0.75, the degree 5 polynomial gives 0.6817; computed absolute error is 2.628e-5. Error is at most 2.649 × 10^-5.

Polynomial value0.6817

Computed absolute error2.628e-5

Guaranteed upper bound2.649 × 10^-5

At x = 0.75, the degree 5 polynomial gives 0.6817; computed absolute error is 2.628e-5. Error is at most 2.649 × 10^-5.

Compare the numbers

Computed values for this scenario (rounded for display)
Input x	Function	Polynomial	Computed error	Series scope
0.75	0.6816	0.6817	2.628e-5	Converges
0	0	0	0	Converges
1	0.8415	0.8417	0.000196	Converges
2	0.9093	0.9333	0.024036	Converges
3	0.1411	0.525	0.38388	Converges

What’s going on? The sine series converges for every real x. With m nonzero terms, the highest degree is 2m − 1. Include the zero coefficient of degree 2m and apply Taylor’s theorem: all derivatives of sine have absolute value at most 1, so the error is at most |x|^(2m+1)/(2m+1)!.

Keep in mind. A bound can be much larger than the actual error. Values use ordinary floating-point arithmetic; roundoff can dominate extremely small computed differences. Bounds concern exact mathematical values and are rounded upward for display. Curves are clipped to the graph window; the table still gives numerical values. A finite polynomial does not prove convergence of the infinite series.

Calculator workspace and limits

For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.

Scientific calculator

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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.

Put it to work

Unit 10 problem 1 · No calculator · analytic
Unit 10 problem 2 · No calculator · graphical

For the teacher

Label aₙ and Sₙ on every graph. Ask what a plotted point measures before discussing a limit.

Unit planning and formative guidance →

Glossary and deeper reading · 3 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Sequence
A sequence is an ordered list of terms indexed by integers. A rule can specify each term directly or recursively from earlier terms. For example, aₙ=1/n produces 1, 1/2, 1/3, and so on. An infinite sequence converges when its terms approach a finite limit as the index grows without bound.
Read more: Original source · Sequences →
Partial sum
A partial sum adds the first finitely many terms of a series. The nth partial sum is Sₙ=a₁+a₂+⋯+aₙ when indexing begins at 1. For example, the first three terms of 1/2+1/4+1/8+⋯ give partial sum 7/8. Convergence of a series concerns the limit of these sums, not just the limit of the individual terms.
Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
Read more: 1.2 · Limit →

Let's work through it

Try it from memory

Try it before you peek.

Question 1

Retrieval 1

Retrieval 2

Catch the tempting mistake

Make the picture move

A polynomial is local evidence, not a universal replacement.

Put it to work

Sequence

Partial sum

Limit

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