In a geometric sequence each term is a constant multiple of the previous term (the multiple here is 2). The general form of a geometric sequence is
sn =
s1rn - 1
where
r is the common ratio (i.e., the amount that each term is multiplied by to get the next term). Obviously,
r = 1 and
r = 0 are not useful cases (both just give a constant value for all terms). It is clear from the graph that the example sequence is divergent, because the terms keep getting bigger.
3. Another geometric sequence
Select the third example, showing another geometric sequence with a common ratio of 1/2. Does this one converge? The terms get closer and closer to zero, so this sequence does converge. Geometric sequences converge if the common ratio is between 0 and 1, and diverge if the common ratio is greater than 1.
4. Alternating geometric sequence
Select the fourth example, showing another geometric sequence with a negative common ratio. Note that the terms alternate on the positive and negative side of the axis. This sequence also converges towards 0, so we can extend our knowledge of geometric sequence convergence to say that the sequence converges if |r| < 1.
Explore
You can experiment with your own sequences by typing in a rule, using n as the variable.
Other 'Sequences and Series' topics
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