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11.2

11.2. Series. Sequences and Series. A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms. Infinite sequences and series continue indefinitely. Infinite Series. In general, if we try to add the terms of an infinite sequence

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11.2

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  1. 11.2 • Series

  2. Sequences and Series • A series is the sum of the terms of a sequence. • Finite sequences and series have defined first and last terms. • Infinite sequences and series continue indefinitely.

  3. Infinite Series • In general, if we try to add the terms of an infinite sequence • we get an expression of the form • a1 + a2 + a3 + . . . + an + . . . • which is called an infinite series (or just a series) and is denoted, for short, by the symbol

  4. nth Partial Sum of an Infinite Series • We consider the partial sums • s1 = a1 • s2 = a1 + a2 • s3 = a1 + a2 +a3 • s4 = a1 + a2 + a3 + a4 • and, in general, • sn =a1 + a2 + a3 + . . . + an= • These partial sums form a new sequence {sn}, which may or may not have a limit.

  5. Definitions • If limnsn = s exists (as a finite number), then we call it the sum of the infinite series: an.

  6. Special series: Geometric Series • A geometric Series is the sum of the terms of a Geometric Sequence: • a +ar +ar2 + ar3 + . . . + arn–1 +. . . = a 0 • Each term is obtained from the preceding one by multiplying it by the common ratior.

  7. Geometric series - Examples: • Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, and finance. • Common ratio: the ratio of successive terms in the series

  8. Geometric series: nth partial sum • If r 1, we have • sn = a +ar + ar2 + . . . + arn-1 • and rsn = ar + ar2 + . . . + arn-1 + arn • Subtracting these equations, we get • sn– rsn = a– arn

  9. Geometric series: limit of the nth partial sum • If –1< r < 1, we know that as rn 0 as n , When | r | < 1 the geometric series is convergent and its sum is: a/(1 – r ). • If r  –1 or r > 1, the sequence {rn} is divergent and so • limnsn is infinite. The geometric series diverges in • those cases.

  10. Summary: Geometric series • When r=-1, the series is called Grandi's series, and is divergent.

  11. All Series: Convergence and Divergence • The converse of this Theorem is not true in general. • If limnan = 0, we cannot conclude that an is convergent!

  12. Series: Test for Divergence • The Test for Divergence follows from Theorem 6 because, if the series is not divergent, then it is convergent, and so • limnan = 0.

  13. Analogy between series and functions:

  14. Practice:

  15. Answers:

  16. Application of geometric series: Repeating decimals

  17. Special case: Telescoping series

  18. A term will cancel with a term that is farther down the list. • It’s not always obvious if a series is telescoping or not until you try to get the partial sums and then see if they are in fact telescoping.

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