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Intro to Sequences and Series One day they decide to go camping in FarmVille!!! They are enjoying the camp fire and the turtle starts to tell a story. He says: “This is a real story about my great great great great great great ….. grandfather…….This is called Zeno’s paradox. ……..” The duckling is to tired to listen to the whole story and falls asleep!!! Zzzzzzz! Shoot! Zeno’s paradox 1 km 1/2 1/4 1/8 1 1 1 1 1 , , , , ......... 2 4 8 16 32 1 2 1 a2 4 1 a3 8 1 a4 16 a1 General term ak 1 2k This is called a sequence. Informally a sequence is an infinite list. What is a sequence of real numbers? More formally… Input Output 1 A sequence of real 1st 2 numbers is a function nd 1 2 in which the inputs are 4 1 rd positive integers and the 3 8 outputs are real numbers. 1 4th 16 I have to walk all these pieces, but……. 1 1 1 1 1 ......... 2 4 8 16 32 This is called an infinite series. To save some time how can I write this sum? 1 k k 1 2 Would this ever end? Namely does this sum has a finite value? Geometrically… 1 2 1 4 1 1 8 1 16 1 1 1 32 1 ……………. 1 64 1 1 To find the total distance that the duckling needs to walk, we add up all the areas… + + + + ……………. • What are these rectangles trying to do? Riemann approximation • For which integrand? For which integral? f ( x) 0 1 2x 1 dx 2x • Is this approximation an over or underestimate? Underestimate What do you know about the integral 1 0 2 x dx ? Is it convergent or divergent? t 1 1 1 dx lim dx lim 0 2 x t 0 2 x t (ln 2)2 x So, it is convergent, namely 1 1 0 2 x dx ln 2 1 1 1 lim ln 2 0 t (ln 2)2t ln 2 t Conclusion: Since the sum of the areas of the rectangles are smaller than the area A below the graph of than 1 ln 2 1 2x , these areas add up to a finite number that is less . 1 1 1 1 1 ........ 2 4 8 16 ln 2 1 1 ln 2 The concepts that the duckling has learned: • Sequences A general sequence a1 , a2 , a3 , a4 ,.... can be written more compactly as ak k=1 or simply ak . •Infinite series a k 1 k •How they can be connected to integrals, convergence, divergence ideas… •Don’t mess with infinity!!! THE END Calculus is awesome! I am happy!