# Geometric series def a geometric series is one that

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Geometric Series: Def:A geometric series is one that can be written in the form n23n0araararar...==++++Note: a=1stterm of the series and r=the ratio of any two consecutive terms.
Page 83 Geometric Series Test (GST): A geometric series converges if ____________ and diverges if _____________. If r1=then nn 0n 0ara====If r1= −then nnn 0n 0ara( 1)====Suppose | r |1. We now find a formula for nSand compute nnlimS→to find nn0Sar==nnlimr→=if1r1if r1if r1 −So if | r |1then S=and if | r |1then S=or Sum of a Geometric Series: If nn 0ar=converges then nn 0ar== In Examples 1 to 5, Show whether the series converges or diverges. If it converges, find the sum. Example 1: nn0110=Example 2: nn02134=
Page 84 Example 3: nn01423=Example 4: 12421...+ −+Properties of Series: If naand nbboth converge, (1) ()nnnnabab=and (2) nncaca=Example 5: nnnn 16223=Telescoping Series:Def:A telescoping series is one in which for any given number of terms, all terms cancel out except for a few terms at the beginning of the series and a few terms at the end.
Page 85 For Example 6 and 7, find the sum Example 6: n 188nn1=+Example 7: 2n26n1=Some Logic: Let p and q be statements. Suppose “If pthen qis true. For example, consider the true statement “If it is Saturday, then it is the weekend.” Write the following statements and decide if they are true or false. If q then p If __________________________ then __________________________If not p then not q If ___________________________ then __________________________ If not q then not p If ___________________________ then _________________________ Theorem: If nn 1a=converges, thennnlim a0→=. What would be another true statement we can deduce from this theorem?
Page 86 The Nth Term Test (NTT) Ifnnlim a0→thennn 1a=diverges For Example 8 and 9, determine whether the series converges or diverges. Example 8: n 1n8n4=+Example 9: nn 1e=Example 10: A ball dropped from a height of 8 ft. rebounds to 75% of its height. Find the total distance the ball travels after bouncing to a stop.
Page 87 Section 9.3 The Integral Test and P-Series: The Integral Test (IT): If f(x)is positive, continuous, and decreasing for x1and naf(n)=then nn 1a=and 1f (x)dxeither both converge or both diverge. If 1f (x)dxdiverges to infinity, then nn 1a=diverges since it’s ________________________________________. If 1f (x)dxconverges to a real number L, then nn 2a=converges since it’s ______________________________. If nn 2a=converges then so does nn 1a=. Why? Note: Because convergence or divergence of a series is not affected by deleting the first N terms, you can use the Integral Test if the conditions are satisfied for all xNfor some integer N1Example 1: Consider n 11n=(Called the harmonic series) Does it converge or diverge? Apply the Nth Term Test: Classical Method: n 111111111111111111...
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