Exercises
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.
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.
Study the series defined by
ak =
k/(k4 + 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 + q2 +
q3 + q4 + ... = 1/(1 - q)
for q=1/3.
Try, before entering anything,
to estimate whether the series defined by
ak = 1/kk
- converges slowly,
- converges fast, or
- diverges.
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).