Exercises

Find an expression for a_{k}
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.

Find an expression for a_{k}
which gives rise to the following sequence of partial sums:
(1,
0,
1,
0,
1,
0 ...)
If you are unsure, just try some variants.

Study the series defined by
a_{k} =
k/(k^{4} + 1).
Do the numerical values suggest convergence?
Confirm your answer by a mathematically exact argument.

Check  at some reasonable accuracy 
the formula for the geometric series
(being valid for all
q<1)
1 + q + q^{2} +
q^{3} + q^{4} + ... = 1/(1  q)
for q=1/3.

Try, before entering anything,
to estimate whether the series defined by
a_{k} = 1/k^{k}
 converges slowly,
 converges fast, or
 diverges.

Insert the expression
4*(1)^k/(2*k+1)
into the text field for a_{k}
and compute the
500^{th} and the
501^{st} partial sum
(each sum starting with
k=0).
Thereby, use a large stepwidth 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
500^{th} and the
501^{st} partial sum
(e.g. using the
maths online minicalculator).