Sequences and series describe patterns that continue term by term, often without a final endpoint. A sequence is an ordered list of numbers, while a series is the sum formed by adding those numbers. These ideas are central in calculus because they let us approximate complicated functions, model long term behavior, and understand infinity in a precise way.
They also connect algebraic patterns to limits, convergence, and real world accumulation.
Understanding Series and Sequences
The important object for an infinite sum is not an impossible act of adding forever. It is the running total after one term, two terms, three terms, and so on. Each running total is a partial sum.
Calculus studies the limit of those totals. A sum can settle even though more positive amount keeps being added, because the added amounts can become tiny fast enough.
For example, repeatedly taking half of what remains produces totals that get closer to a fixed amount. This is why infinity in calculus means a limiting process, not a very large final number.
It is essential to keep the behavior of terms separate from the behavior of sums. Terms approaching zero only clear one necessary condition for a sum to converge. They do not prove convergence.
The harmonic series makes this point clearly. Its terms become small, yet they shrink too slowly, so the running total keeps growing. Grouping terms helps reveal why.
After the first few terms, groups can contain twice as many terms as the previous group. Even though every term in a later group is small, the whole group still adds a noticeable amount. This creates endless growth.
Geometric series work because every new term is made by multiplying by the same ratio. When the ratio has size less than one, the terms shrink by a fixed percentage each step. The remaining amount becomes smaller in a very controlled way.
This pattern appears in bouncing balls, repeated discounts, radioactive decay measured at equal time intervals, and digital images that use repeated scaling. A negative ratio causes terms to alternate above and below zero.
The partial sums may zigzag, but the zigzags get narrower when the ratio has size less than one. If the ratio has size one or greater, the terms fail to shrink enough and the total cannot settle.
Convergence tests are tools for judging sums whose exact total may be hard or impossible to find. Comparison tests use a familiar series as a benchmark. If positive terms are no larger than terms from a known convergent series, their sum converges.
If they are no smaller than terms from a known divergent series, their sum diverges. The ratio test is especially useful when factorials or powers appear, since it examines how one term compares with the next.
Students should first check the term limit before using a more complicated test. They should then state what test applies, verify its conditions, and distinguish a conclusion about convergence from a calculation of the actual sum.
Key Facts
- A sequence is a function of the positive integers: a_n gives the nth term.
- A series is the sum of a sequence: S = a_1 + a_2 + a_3 + ... = sum from n = 1 to infinity of a_n.
- A sequence converges to L if lim as n approaches infinity of a_n = L.
- A series converges if its partial sums S_N = sum from n = 1 to N of a_n approach a finite limit.
- Geometric series test: sum from n = 0 to infinity of ar^n = a/(1 - r) when |r| < 1.
- nth term test for divergence: if lim as n approaches infinity of a_n is not 0, then sum a_n diverges.
Vocabulary
- Sequence
- A sequence is an ordered list of terms, usually written a_1, a_2, a_3, and so on.
- Series
- A series is the sum of the terms of a sequence.
- Partial sum
- A partial sum is the sum of the first N terms of a series, written S_N.
- Convergence
- Convergence means that a sequence or the partial sums of a series approach a finite value.
- Divergence
- Divergence means that a sequence or series does not approach a finite limit.
Common Mistakes to Avoid
- Confusing a sequence with a series is wrong because a sequence lists terms, while a series adds them.
- Using the nth term test to prove convergence is wrong because the test can only prove divergence when the term limit is not zero.
- Assuming a_n approaches 0 means sum a_n converges is wrong because the harmonic series has terms that approach 0 but still diverges.
- Forgetting to check |r| < 1 in a geometric series is wrong because the formula a/(1 - r) only applies when the infinite sum converges.
Practice Questions
- 1 Find the first 5 terms of the sequence a_n = 3n - 2, and then find the partial sum S_5.
- 2 Determine whether the geometric series sum from n = 0 to infinity of 5(0.4)^n converges, and if it does, find its sum.
- 3 Explain why the series sum from n = 1 to infinity of 1/n cannot be shown to converge just because its terms approach 0.