9.2 Geometric Series Review
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1 9.2 Geometric Series Review Geometric Series a= starting term, x=constant ratio of each term to preceding one ax i = i=0 a x when x < and diverges otherwise n th partial sum (st n terms added): n ax i = a( x n ) x for x Example: n 2 2 i=0 i = n i= i=0 i 2 careful of starting # and index Math 20: Calculus and Analytic Geometry II
2 9.3 Series: Partial Sums More Generally a n and convergence? [9.3, 9.4, 9.5, chapter 0] n= Math 20: Calculus and Analytic Geometry II
3 9.3 Series: Partial Sums More Generally a n and convergence? [9.3, 9.4, 9.5, chapter 0] n= n th partial sum (st n terms added): n i= a i = n a i i=0 sequence of partial sums S n converges series does so examine lim S n n Example: n= n(n+) Math 20: Calculus and Analytic Geometry II
4 9.3 Series: Partial Sums More Generally a n and convergence? [9.3, 9.4, 9.5, chapter 0] n= n th partial sum (st n terms added): n i= a i = n a i i=0 sequence of partial sums S n converges series does so examine lim S n n Example: n(n+) S n = n n+ n= Math 20: Calculus and Analytic Geometry II
5 9.3 Series: Partial Sums More Generally a n and convergence? [9.3, 9.4, 9.5, chapter 0] n= n th partial sum (st n terms added): n= n i= a i = n a i i=0 sequence of partial sums S n converges series does so examine lim S n n Example: n(n+) S n = n n+ lim S n = n Math 20: Calculus and Analytic Geometry II
6 9.3: Terms Not Going to 0 terms not going to 0: lim a n 0 or DNE, then partial n sums diverge and so does the series. Example: n= 5+n 2n+ Math 20: Calculus and Analytic Geometry II
7 Clicker Question. What can we say about the series () n? n= a) it is a geometric series with a constant ratio of each term to its preceding one x b) we can find a pattern for the partial sums S n = n a i i= c) lim n a n 0 so we can apply the terms not going to 0 d) all of the above Math 20: Calculus and Analytic Geometry II
8 Clicker Question. What can we say about the series () n? n= a) it is a geometric series with a constant ratio of each term to its preceding one x b) we can find a pattern for the partial sums S n = n a i i= c) lim n a n 0 so we can apply the terms not going to 0 d) all of the above Math 20: Calculus and Analytic Geometry II
9 9.3:Linearity for Convergence or Divergence Linearity: a n converges to S and b n converges to n= T, and k is any constant, then ks + T. n= ka n + b n converges to n= Math 20: Calculus and Analytic Geometry II
10 9.3:Linearity for Convergence or Divergence Linearity: a n converges to S and n= T, and k is any constant, then b n converges to n= ka n + b n converges to n= ks + T. Application : add two geometric series (converge to sum) Math 20: Calculus and Analytic Geometry II
11 9.3:Linearity for Convergence or Divergence Linearity: a n converges to S and n= T, and k is any constant, then b n converges to n= ka n + b n converges to n= ks + T. Application : add two geometric series (converge to sum) Application 2: add divergent & convergent series (diverge) Example: ( 2 )n + n. n= Diverges, because if it were convergent, then subtract convergent ( 2 )n and the result should converge by n= linearity, but doesn t! Math 20: Calculus and Analytic Geometry II
12 2. What can we say about Clicker Question n= 3 n + 2 n? a) It is a geometric series so we can apply 9.2 methods to determine convergence by checking if x < or divergence otherwise b) We can use the lim n a n 0 to determine divergence c) We can use linearity to determine convergence or divergence d) all of the above e) none of the above Math 20: Calculus and Analytic Geometry II
13 2. What can we say about Clicker Question n= 3 n + 2 n? a) It is a geometric series so we can apply 9.2 methods to determine convergence by checking if x < or divergence otherwise b) We can use the lim n a n 0 to determine divergence c) We can use linearity to determine convergence or divergence d) all of the above e) none of the above = n 3 n + n 2 n = ( 3 )n + ( 2 )n n= n= Math 20: Calculus and Analytic Geometry II
14 Clicker Question 3. Do any of the following apply to n= n? a) It is a geometric series so we can apply 9.2 methods to determine convergence by checking if x < or divergence otherwise b) We can use the lim n n 0 to determine divergence of c) We can use linearity to determine convergence d) all of the above e) none of the above n= n Math 20: Calculus and Analytic Geometry II
15 Clicker Question 3. Do any of the following apply to n= n? a) It is a geometric series so we can apply 9.2 methods to determine convergence by checking if x < or divergence otherwise b) We can use the lim n n 0 to determine divergence of c) We can use linearity to determine convergence d) all of the above e) none of the above n= n Math 20: Calculus and Analytic Geometry II
16 Math 20: Calculus and Analytic Geometry II
17 Harmonic series N= N diverges by growing to slowly! Why? Integral Test Math 20: Calculus and Analytic Geometry II
18 9.3: Integral Test Bounds If series has terms that are decreasing and positive (eventually), the integral test not only tells us about convergence, but also bounds the series: a =f() a a 2 a 3 a 4 Math 20: Calculus and Analytic Geometry II
19 9.3: Integral Test Bounds If series has terms that are decreasing and positive (eventually), the integral test not only tells us about convergence, but also bounds the series: a =f() a a 2 a 3 a 4 f (x)dx a n Math 20: Calculus and Analytic Geometry II
20 9.3: Integral Test Bounds If series has terms that are decreasing and positive, the integral test not only tells us about convergence, but also bounds the series: a =f() a a 2 a 3 a 4 Math 20: Calculus and Analytic Geometry II
21 9.3: Integral Test Bounds If series has terms that are decreasing and positive, the integral test not only tells us about convergence, but also bounds the series: a =f() a a 2 a 3 a 4 a a 2 a 3 a 4 a 5 f (x)dx a n a + f (x)dx Math 20: Calculus and Analytic Geometry II
22 Harmonic series N= N diverges by growing to slowly! Why? Integral Test assumptions: terms are decreasing, a n > 0, known integral Math 20: Calculus and Analytic Geometry II
23 Harmonic series N= N diverges by growing to slowly! Why? Integral Test assumptions: terms are decreasing, a n > 0, known integral yes so test applies Math 20: Calculus and Analytic Geometry II
24 Harmonic series N= N diverges by growing to slowly! Why? Integral Test assumptions: terms are decreasing, a n > 0, known integral yes so test applies b dx = lim x b x dx = Math 20: Calculus and Analytic Geometry II
25 Harmonic series N= N diverges by growing to slowly! Why? Integral Test assumptions: terms are decreasing, a n > 0, known integral yes so test applies b dx = lim x b x dx = lim ln(x) b b = Math 20: Calculus and Analytic Geometry II
26 Harmonic series N= N diverges by growing to slowly! Why? Integral Test assumptions: terms are decreasing, a n > 0, known integral yes so test applies b dx = lim x b x dx = lim ln(x) b b = lim ln(b) ln() b diverges so series does too Math 20: Calculus and Analytic Geometry II
27 9.3: Integral Test For a n, if the terms are decreasing and a n > 0 then the series behaves the same way as a n dn. So look for decreasing and positive terms (eventually) that we can integrate (Calc I or Chap 7) + improper integral. Otherwise the test does NOT help. Math 20: Calculus and Analytic Geometry II
28 9.3: Integral Test For a n, if the terms are decreasing and a n > 0 then the series behaves the same way as a n dn. So look for decreasing and positive terms (eventually) that we can integrate (Calc I or Chap 7) + improper integral. Otherwise the test does NOT help. p series conv if p > and div if p by int test n= n p Math 20: Calculus and Analytic Geometry II
29 Math 20: Calculus and Analytic Geometry II
30 Geo series n 2 2 i=0 i = n i= i 2 converges as n to 2 2 slowly (Zeno s paradox) = hamster Math 20: Calculus and Analytic Geometry II
31 Clicker Question 4. Which of the following are true regarding n=2 2n 4+n 2? a) It is a geometric series so we can apply 9.2 methods to determine convergence by checking if x < or divergence otherwise 2n b) lim 0 determines divergence of n 4+n 2 n=2 2n 4+n 2 c) We can use linearity to determine convergence d) We can use the integral test to determine convergence e) none of the above Math 20: Calculus and Analytic Geometry II
32 Clicker Question 4. Which of the following are true regarding n=2 2n 4+n 2? a) It is a geometric series so we can apply 9.2 methods to determine convergence by checking if x < or divergence otherwise 2n b) lim 0 determines divergence of n 4+n 2 n=2 2n 4+n 2 c) We can use linearity to determine convergence d) We can use the integral test to determine convergence e) none of the above Math 20: Calculus and Analytic Geometry II
33 5. Does the series Clicker Question ( ) n converge? n= a) yes and I have a good reason why b) yes but I am unsure of why c) no, but I am unsure of why d) no, and I have a good reason why e) it is not a series, so no Math 20: Calculus and Analytic Geometry II
34 5. Does the series Clicker Question ( ) n converge? n= a) yes and I have a good reason why b) yes but I am unsure of why c) no, but I am unsure of why d) no, and I have a good reason why e) it is not a series, so no Math 20: Calculus and Analytic Geometry II
35 History and Applications Brahmagupta gave rules for summing series in his 628 work Brahmasphutasiddanta (Opening of the Universe) wheat (or rice) and chess problem. Stories of 63 n=0 2 n grains owed by King (8,446,744,073,709,55,65) Nicole Oresme (4th century) harmonic series diverging. Name from wavelengths of the overtones of a vibrating string. Architects. James Gregory (668) introduced the terms convergence and divergence Integral test was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin-Cauchy test (or by either name). infinite series are widely used in mathematics & other quantitative disciplines such as physics, computer science, & finance. Math 20: Calculus and Analytic Geometry II
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