Exercises
  1. Find an expression for an which describes the series

    (10, 20, 30, 40, 50, 60, 70 ...)

    If you are unsure, just try some variants.
     
  2. Find an expression for an which describes the series

    (0.1, 0.01, 0.001, 0.0001 ...)

    If you are unsure, just try some variants.
     
  3. Study the sequence given by an = (n + 3)/(n + 4). Do the numerical values suggest that it converges? Which number appears to be the limit? Increase the upper bound (and step-width), in order to illustrate the situation for large values of n.
    Confirm your results by a mathematically exact argument.
     
  4. The sequence defined by an = 2 + 1/n - 1/(n+1) converges against 2. Due to the exact definition of convergence, for any e > 0, there is an item in the sequence, such that for it and for all following items the inequality

    | an - 2 | < e

    holds. Use the program in order to determine, from which item the inequality | an - 2 | < 1/1000 is valid.
     
  5. In case you know what the sine function is: Use the program to study the sequences given by an = sin(n) and bn = sin(n)/n. How do they behave for large n?
     
  6. For friends of programming: Type (or copy-and-paste) the following JavaScript code into the text field for an:
       function f(x) {if (x==1) return 1
       else if (x==2) return 1
       else return f(x-1)+f(x-2)}
       f(n)
    
    Which sequence is described by this? (Initial value: n = 1).