Sequences and Series
Convergence, Tests & Sums
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Sequences and series are tools for describing what happens when a process continues step by step forever. A sequence lists numbers in order, while a series adds those numbers together. Convergence tells us whether the terms of a sequence approach a single value or whether the partial sums of a series settle toward a finite number. This idea matters because infinite processes appear in calculus, physics, engineering, and numerical computation.
To study convergence, mathematicians compare terms or partial sums to a limit and ask whether they get arbitrarily close after enough steps. For sequences, the key question is whether approaches as becomes large. For series, we examine the sequence of partial sums and test whether approaches a finite value. Different convergence tests, such as the geometric series rule, comparison test, ratio test, and integral test, help determine what happens without adding infinitely many terms directly.
Key Facts
- A sequence converges to L if for every epsilon > 0, there exists N such that n > N implies |a_n - L| < epsilon.
- A series converges if its partial sums approach a finite limit.
- If converges, then . The converse is false.
- Geometric series: converges to when .
- p-series: converges if and diverges if .
- Ratio test: if , then the series converges for and diverges for .
Vocabulary
- Sequence
- A sequence is an ordered list of numbers written as a_1, a_2, a_3, and so on.
- Series
- A series is the sum of the terms of a sequence, often written as a_1 + a_2 + a_3 + ...
- Limit
- A limit is the value that sequence terms or partial sums approach as the index grows without bound.
- Partial sum
- A partial sum is the sum of the first n terms of a series, usually denoted S_n.
- Convergence test
- A convergence test is a method used to decide whether an infinite series converges or diverges.
Common Mistakes to Avoid
- Assuming guarantees that converges, but this is wrong because many series, such as , have terms going to zero and still diverge.
- Applying a sequence limit directly to a series, which is wrong because a series must be studied through its partial sums rather than through the terms alone.
- Using the geometric series formula when |r| >= 1, which is wrong because the infinite geometric series only converges for |r| < 1.
- Stopping after the ratio test gives r = 1, which is wrong because the ratio test is inconclusive there and another test must be used.
Practice Questions
- 1 Determine whether the sequence a_n = (3n + 1)/(n + 5) converges, and if it does, find its limit.
- 2 Decide whether the series converges, and if so, find its sum.
- 3 A student says that because the terms of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... get very small, the series must converge. Explain clearly why this reasoning is not sufficient.