Exercises
  1. Find an expression for ak which gives rise to the following sequence of partial sums:

    (1, 2, 3, 4, 5, 6, 7 ...)

    If you are unsure, just try some variants.
     
  2. Find an expression for ak which gives rise to the following sequence of partial sums:

    (1, 0, 1, 0, 1, 0 ...)

    If you are unsure, just try some variants.
     
  3. Study the series defined by ak = k/(k4 + 1). Do the numerical values suggest convergence? Confirm your answer by a mathematically exact argument.
     
  4. Check - at some reasonable accuracy - the formula for the geometric series (being valid for all |q|<1)

    1 + q + q2 + q3 + q4 + ... = 1/(1 - q)

    for q=1/3.
     
  5. Try, before entering anything, to estimate whether the series defined by ak = 1/kk

        -   converges slowly,
        -   converges fast, or
        -   diverges.
     
  6. Insert the expression
    4*(-1)^k/(2*k+1)
    
    into the text field for ak and compute the 500th and the 501st partial sum (each sum starting with k=0). Thereby, use a large step-width in order to avoid a long output list. Does the series converge to a number you know? In order to increase the accuracy, form the mean value of the 500th and the 501st partial sum (e.g. using the maths online mini-calculator).