Convergence of Taylor Series

Part 2

J. Gonzalez-Zugasti, University of 1 Massachusetts - Lowell Using Taylor Series

Since every Taylor series is a power series, the operations of adding, subtracting, and multiplying Taylor series are all valid on the intersection of their intervals of convergence.

J. Gonzalez-Zugasti, University of 2 Massachusetts - Lowell Example 1

Find the Taylor series at = 0 of . −6푥 Solution: 푥 푒 Since

= ∞ 푘! 푥 푥 we get 푒 � 푘=0 푘

= ∞ ! 푘 −6푥 −6푥

푒 J. Gonzalez-Zugasti,� University of 3 Massachusetts푘=0 - Lowell푘 Example 2

Find the Taylor series at = 0 of . 푥 Solution: 푥 3푥푒 Since

= ∞ 푘! 푥 푥 we get 푒 � 푘=0 푘

= ∞ = ∞ 푘! 푘!+1 푥 푥 3푥

3푥푒 3푥J. Gonzalez�-Zugasti, University� of 4 Massachusetts푘=0 푘 - Lowell 푘=0 푘 Example 3

Find the Taylor series for cos at = 0. 휋푥 2 푥 2 푥 Solution: Since

cos = 1 ∞ 2푘 ! 푘 푥 푥 � − we get 푘=0 2푘

J. Gonzalez-Zugasti, University of 5 Massachusetts - Lowell Example 3 (continued)

2 cos = 1 2푘 2 ∞ 휋� ! 휋� 푘 � − 푘=0 2푘 = ∞ 1 2 2푘 2푘 ! 푘 휋 푥 Therefore � − 푘=0 2푘

cos = ∞ 1 2 2 2푘 2푘 ! 2 휋� 2 푘 휋 푥 푥 푥 � − 푘=0 2푘 = ∞ 1 2 2푘 2푘+2! 푘 휋 푥 � − 푘=0 2푘 J. Gonzalez-Zugasti, University of 6 Massachusetts - Lowell Example 4

Find the Taylor series at = 0 of . 2 푥 Solution: 푥 1−푥 Since 1 = ∞ 1 푘 we get � 푥 − 푥 푘=0

= ∞ = ∞ 1 2 푥 2 푘 푘+2 푥 � 푥 � 푥 J. Gonzalez-Zugasti, University of 7 − 푥 Massachusetts푘=0 - Lowell 푘=0 Example 5

Find the first four nonzero terms in the Maclaurin series for = sin . −6푥 Solution: 푓 푥 푒 푥 We will use the Taylor series for and sin and then use: If and converge absolutely−6푥 for < and 푛 푛 푒 푥 푛 푛 ∑ 푎 푥 ∑ 푏 푥 = 푛 푥 푅 then 푐푛 � 푎푘푏푛−푘 푘=0 = 푛 푛 푛 which also converges� for푎푛 푥 < �. 푏푛푥 � 푐푛푥

J. Gonzalez-Zugasti, University of 8 푥 Massachusetts푅 - Lowell Example 5 (continued)

= = 1 + 18 36 + 54 ∞ ! 푘 −6푥 −6푥 2 3 4 So푒 we will� let = 1, =− 6푥6, =푥18−, =푥 36, 푥 =− 54⋯ . 푘=0 푘

푎0 푎1 − 푎2 푎3 − 푎4 sin = 1 ∞ 2푘++1 ! 푘 푥 푥 � − 1 =푘=00+ + 02푘 + 0 + 6 2 3 4 So we will let 푥 = 0∙,푥 −= 1,푥 = 0∙,푥 = ⋯ , = 0. 1 푏0 푏1 푏2 푏3 − 6 푏4 J. Gonzalez-Zugasti, University of 9 Massachusetts - Lowell Example 5 (continued)

= 푛

= 푐푛 � 푎푘푏푛−푘 푘=0 = 1 0 = 0 푐0 = 푎0푏0 + = 1 ∙ 1 + 6 0 = 1 푐1 = 푎0푏1 + 푎1푏0 + = 1 ∙ 0 + −6 ∙ 1 + 18 0 = 6 푐2 = 푎0푏2 + 푎1푏1 + 푎2푏0 + ∙ 1− ∙ ∙ − 107 3 =01 3 1 2+ 26 1 0 +3180 1 + 36 0 = 푐 푎 푏 푎6푏 푎 푏 푎 푏 6 = ∙ +− + − ∙+ +∙ − ∙ 1 4 =01 4 0 + 1 36 2 2 + 183 1 0 +4 036 1 + 54 0 = 35 푐 푎 푏 푎 푏 푎 푏6 푎 푏 푎 푏 ∙ − − ∙ − ∙ ∙ −

J. Gonzalez-Zugasti, University of 10 Massachusetts - Lowell Example 5 (continued)

Therefore

sin = 1 ∞ ! 푘 ∞ 2푘++1 ! −6푥 −6푥 푘 푥 푒 푥 � � − 푘=0 푘 푘=0 2푘 = ∞ 푛 � 푐푛푥 107 = 푘=0 6 + 35 + 6 2 3 4 푥 − 푥 푥 − 푥 ⋯

J. Gonzalez-Zugasti, University of 11 Massachusetts - Lowell Example 6

Find the first four nonzero terms in the Maclaurin series for . 3 2 1+푥 Solution: Since 1 = 1 ∞ 푘 we get � 푥 − 푥 푘=0 1 = = 1 1 + ∞ ∞ 푘 푘 푘 � −푥 � − 푥 J. Gonzalez-Zugasti, University of 푥 푘=0 푘=0 12 Massachusetts - Lowell Example 6 (continued)

Notice that 1 1 = 1 + 1 + 푑 − and that 2 푑� 푥 푥 1 = 1 1 + ∞ 푑 푑 푘 푘 � − 푥 푑� 푥 푑� 푘=0 = ∞ 1 푑 푘 푘 � − 푥 푘=0 푑� = ∞ 1 푘 푘−1 � − 푘푥 푘=0 J. Gonzalez-Zugasti, University of 13 Massachusetts - Lowell Example 6 (continued)

Using these results we get: 3 1 = 3 1 + 1 + − 2 − 2 푥= 3 ∞ 1 푥 푘 푘−1 − � − 푘푥 푘=0 = ∞ 1 3 푘+1 푘−1 � − 푘푥 푘=0 = 3 + 9 12 + 2 3 J. Gonzalez-Zugasti, University of 14 − 6푥 푥 −Massachusetts푥 - Lowell ⋯ Why You Need to Know Taylor Series

Russian physicist Igor Tamm won the Nobel Prize in physics in 1958. During the Russian revolution, he was a physics professor at the University of Odessa in the Ukraine. Food was in short supply, so he made a trip to a nearby village in search of food. While he was in the village, a bunch of anti-communist bandits surrounded the town. The leader was suspicious of Tamm, who was dressed in city clothes. He demanded to know what Tamm did for a living. He explained that he was a university professor looking for food. “What subject?,” the bandit leader asked. Tamm replied “I teach mathematics.” “Mathematics?” said the leader. “OK. Then give me an estimate of the error one makes by cutting off a Maclaurin series expansion at the n th term. Do this and you will go free. Fail, and I will shoot you.” Tamm was not just a little astonished. At gunpoint, he managed to work out the answer. He showed it to the bandit leader, who perused it and then declared “Correct! Go home.” Tamm never discovered the name of the bandit.

From “Calculus makes you live longer”, in “100 essential things you didn’t know you didn’t know”, by John Barrow. http://math.stackexchange.com/questions/28885/anecdotes-about-famous-mathematicians-or-physicists

J. Gonzalez-Zugasti, University of 15 Massachusetts - Lowell