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Sequences and Series
Convergence, Tests & Sums
Related Tools
Related Labs
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 a_n approaches L as n becomes large. For series, we examine the sequence of partial sums S_n = a_1 + a_2 + ... + a_n and test whether S_n 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 sum a_n converges if its partial sums S_n = sum from k=1 to n of a_k approach a finite limit.
- If sum a_n converges, then a_n -> 0. The converse is false.
- Geometric series: sum from n=0 to infinity of ar^n converges to a/(1 - r) when |r| < 1.
- p-series: sum from n=1 to infinity of 1/n^p converges if p > 1 and diverges if p <= 1.
- Ratio test: if lim n->infinity |a_(n+1)/a_n| = r, then the series converges for r < 1 and diverges for r > 1.
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 a_n -> 0 guarantees that sum a_n converges, but this is wrong because many series, such as sum 1/n, 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 sum from n=1 to infinity of 4(1/3)^n 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.