Princeton Review AP Calculus BC, Chapter 12: Infinite Sequences and Series

### Sequences & Series

* **Sequence:** An infinite succession of numbers that follow a pattern
* Usually denoted with a subscript using aₙ
* Usually the terms are generated by a formula, ex. aₙ=(n-1)/n beginning with n=1 is 0, 1/2, 2/3, etc.
* n is always an integer
* Officially:
* “A sequence has a limit L if for any ε>0 there is an associated positive integer *N* such that |aₙ - L| < ε for all *n* ≥ *N.* If so, the sequence converges to *L* and we write lim (n → ∞) aₙ = L”
* If a sequence has no finite limit, it diverges.

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* **Infinite Series**
* An infinite series is an expression of the form a₁ + a₂ + a₃… + aₖ + …. The numbers a₁, a₂, a₃, etc. are the *terms* of the series
* In other words, an infinite series is a sequence of terms where the terms are added up, there is a pattern to the order of the terms, and there are an infinite number of terms.

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* As *n* increases, the partial sum includes more and more terms of the series. So if aₙ approaches a limit as n approaches ∞, then this limit is likely to be the sum of all of the terms in the series, thus a **convergence.**
* If the sequence of partial sums converges to a limit *S,* then the series is said to converge and S is called the sum of the series.
* If the sequence of partial sums diverges (doesn’t have a limit), then the series is said to diverge which means that a divergent series has no sum.

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### **Geometric Series**

* Used in the form of ∞Σ(n=1) a\*r^(n-1)

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* These show up often on the AP exam: often asked to determine whether the series converges or diverges, and if it converges, to what limit.
* If |r|