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Section 9.6 Calculus BC AP/Dual, Revised Β©2018 viet.dang@humbleisd.net
Ratio and Root Test Section 9.6 Calculus BC AP/Dual, Revised Β©2018 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Summary of Tests for Series
Looking at the first few terms of the sequence of partial sums may not help us much so we will learn the following ten tests to determine convergence or divergence: P π-series: Is the series in the form π π π· ? A Alternating series: Does the series alternate? If it does, are the terms getting smaller, and is the πth term 0? R Ratio Test: Does the series contain things that grow very large as π increases (exponentials or factorials)? R Root Test: Does the series contain a radical? T Telescoping series: Will all but a couple of the terms in the series cancel out? I Integral Test: Can you easily integrate the expression that define the series? N πth Term divergence Test: Is the nth term something other than zero? G Geometric series: Is the series of the form, π=π β π π π C Comparison Tests: Is the series almost another kind of series (e.g. π-series or geometric)? Which would be better to use: Direct or Limit Comparison Test? 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Ratio Test Let π=π β π π be a series in which π π >π for all π (or at least all π past some particular threshold value π΅). Form the ratio π π+π π π and evaluate its limit as πββ. Provided this limit exists, there are three possible cases: If π₯π’π¦ πββ π π+π π π <π , then π=π β π π converges If π₯π’π¦ πββ π π+π π π >π , then π=π β π π diverges If π₯π’π¦ πββ π π+π π π =π , the Ratio Test is inconclusive. Then, π=π β π π could either converge or diverge as another test is needed to decide the series. 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 1 Determine whether the following converges or diverges of, π=π β π π π! 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 2 Determine whether the following converges or diverges of, π=π β (βππ) π π ππ+π π+π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 2 Determine whether the following converges or diverges of, π=π β (βππ) π π ππ+π π+π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 3 Determine whether the following converges or diverges of, π=π β π! π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 4 Determine whether the following converges or diverges of, π=π β π+π ππ+π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Your Turn Determine whether the following converges or diverges of, π=π β π π (π π+π ) π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Root Test π=π β π π converges if π₯π’π¦ πββ π π π <π
π=π β π π diverges if π₯π’π¦ πββ π π π >π If π₯π’π¦ πββ π π π =π , the Root Test is inconclusive so another test would be used. 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 5 Determine whether the following converges or diverges of, π=π β π ππ π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 6 Determine whether the following converges or diverges of, π=π β βππ ππ+π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 7 Determine whether the following converges or diverges of, π=π β π π π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 7* Determine whether the following converges or diverges of, π=π β π π π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Example 7* Determine whether the following converges or diverges of, π=π β π π π π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Your Turn Determine whether if the root test converges or diverges of π=π β ππ+π ππ π 12/18/2019 3:12 AM Β§9.6: Ratio and Root Test
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Assignment Page 633 13-31 odd, 35-47 EOO, 51-65 odd 12/18/2019 3:12 AM
Β§9.6: Ratio and Root Test
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