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Warm Up

Warm Up. Write the explicit formula for the series. Evaluate. Introduction to Series. What is a series? What does it mean for a series to converge? What are geometric and telescoping series? What is the nth term test?. Infinite Series. Partial Sums.

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Warm Up

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  1. Warm Up Write the explicit formula for the series. Evaluate.

  2. Introduction to Series What is a series? What does it mean for a series to converge? What are geometric and telescoping series? What is the nth term test?

  3. Infinite Series

  4. Partial Sums

  5. A series can either converge or diverge. If the sequence of nth partial sums converges to A, then the series converges. The limit of S is called the sum of the series. If the sequence of nth partial sums diverges, then the series diverges.

  6. Ex: Find S1, S2, S3, S4, S5, …and an expression for Sn Therefore, the series converges to 1. note: If the limit of the sequence is NOT 0, then the sum of the series must diverge.

  7. This example was a geometric series because an is an exponential function. A geometric series with ratio (base) r • diverges if |r| > 1 • Converges if 0 < r < 1 If it converges then the series converges to

  8. Converge/Diverge? If converge, tell the limit of the series. 1. 2. 3.

  9. Find the sum of the series This is called a telescoping series.

  10. Find the sum of the series This is another telescoping series.

  11. Let’s revisit the “note” we talked about before. • If the limit of the sequence is not 0, then the series diverges. • The contrapositive of this statement must also be true: If the series converges, then the limit of the sequence is 0. • However, the converse (and inverse) do not have to be true…and are NOT in this case. Just because the limit of the sequence is 0, the series can still diverge. • All of this information is classified as the “nth term test for divergence.” Always use this as your first step in answering the converge/diverge question for series.

  12. In Summary: When asked if a series converges or diverges: • Do the nth term test for divergence. • If the series if geometric, find r and determine whether the series converges or diverges. If converges, find the sum. • If the series is a rational function and the denominator can be factored, separate the ratio by partial fractions and determine if it is a telescoping series. If it is, find the value to which it converges.

  13. Test Series Condition(s) for convergence Conditions for divergence Example & Comments

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