5. Sequences

In Sequences we start studying Calculus. Calculus deals with processes beyond the finite. Certainly a fascinating but also a difficult task as we have no direct access to infinity based on our experiences of every day life.

There are many phenomena however that could not be treated properly with only finite arguments. Since Classical Antiquity a lot of paradoxes have played with this insufficiency. The most famous probably is that of Zeno of Elea telling about a speed contest between Achilles and a tortoise.

For the benefit of all we can explain (e.g. here) why Achilles in the end catches up with the tortoise!

Zeno's Paradox

It is impossible for Achilles to win a racing competition with a tortoise as soon as he allows a competitative edge.

Because: To get to the point where the tortoise starts Achilles spends time. During that time however the tortoise moves on, so that when Achilles reaches her starting point the tortoise is no longer there.

  1. Sequences as Special Functions
  2. Recursive Sequences and the Principle of Induction

       Excursus: Binomial Coefficients and Pascal's Triangle

       The Fibonacci Sequence

  3. Monotone Sequences, Bounded Sequences
  4. Convergent Sequences
  5. Properties of Convergent Sequences
  6. Calculation Rules for Convergent Sequences
  7. Monotone and Bounded Sequences

       Babylonian Square Root Algorithm (interactive)

  8. The Bolzano-Weierstrass Theorem
  9. Convergent Series

       Every positive real number x has a g-adic representation

       e is irrational

  10. Double Sequences and Double Series
  11. Convergent Power Series
  12. Analytical Functions