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Intro to Sequences and Series

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.

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Intro to Sequences and Series

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  1. Intro to Sequences and Series

  2. 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 greatgreatgreatgreat ….. grandfather…….This is called Zeno’s paradox. ……..”

  3. The duckling is to tired to listen to the whole story and falls asleep!!! Zzzzzzz!

  4. Shoot! Zeno’s paradox

  5. 1 km

  6. 1/2

  7. 1/4

  8. 1/8

  9. This is called a sequence. Informally a sequence is an infinite list. What is a sequence of real numbers? More formally… Input Output A sequence of real numbers is a function in which the inputs are positive integers and the 3rd outputs are real numbers. 4th 1st 2nd General term

  10. I have to walk all these pieces, but……. This is called an infinite series. To save some time how can I write this sum? Would this ever end? Namely does this sum has a finite value?

  11. Geometrically… ……………. 1 1 1 1 1 1 To find the total distance that the duckling needs to walk, we add up all the areas… + + + + …………….

  12. What are these rectangles trying to do? • Riemann approximation • For which integrand? For which integral? • Is this approximation an over or underestimate? • Underestimate

  13. What do you know about the integral ? Is it convergent or divergent? So, it is convergent, namely

  14. Conclusion: Since the sum of the areas of the rectangles are smaller than the area A below the graph of , these areas add up to a finite number that is less than .

  15. The concepts that the duckling has learned: • Sequences • A general sequence can be written more compactly as • Infinite series • How they can be connected to integrals, convergence, divergence ideas… • Don’t mess with infinity!!!

  16. THE END Calculus is awesome! I am happy!

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