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Calculus
Series and Sequences
Convergence, divergence, and the geometric series
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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.
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.