The members of the geometric sequence follow the rule:
xi+1= xi*a
xi it the i-th member of the sequence, with index i a positive integer, including 0. The growth parameter a is real number. With
x0=1
the members are 1, a, a2, a3, a4..... xi =a i
The geometric series (sequence of partial sums) is generated by consecutive addition of the members of the geometric sequence: The partial sums are:
Si = Σoi a n with 0 ≤ n≤ i; Si = 1 + a + a 2....+a i
In the simulation you can vary parameter a by a slider in the range -1,05≤ a ≤+1,05 . Reset returns to the default value 0.5 .
The interesting issue is if the series will have a finite value (a limit, if the series is convergent) when the index grows unlimited, or if it goes to infinity (the series is divergent). This depends on a.
The left chart displays the members of the sequence, the right chart those of the partial sum series, both in dependence on the index i. In a separate window at the right the chart of the limit of the series of partial sums is shown in dependence on a, with a red point for its chosen value.
When a is smaller than 1 the geometric series of partial sums converges to
limitS i = 1 / (1-a) for abs(a) < 1
The graph in the right window displays this formula for the range-0.98 < a < 0.98
.