The convergence properties of the power series depends on the size of
the base a:
- |a| < 1: the sequence converges to 0.
- a = 1: the sequence converges to 1 (being constant)
- a > 1: the series diverges to plus infinity
- a
-1: the series diverges
Proof:
This seems an obvious statement: if a number is in absolute values less than
one, it gets smaller and smaller when raised to higher and higher powers.
Proving something 'obvious', however, is often difficult, because it may not
be clear how to start. To prove the statement we have to resort to one of
the elementary properties of the real number system: the Archimedian principle.
- Case a > 1:
- Take any real number K > 0 and define
Since a > 1 we know that x > 0. By the Archimedian
principle there exists a positive integer n such that nx > K - 1.
Using Bernoulli's inequality for that n we have:
-
an = (1 + x)n
1 + nx > 1 + (K - 1) = K
But this proves that the sequence {an} is
unbounded, hence it can not converge.
- Case 0 < a < 1:
- Take any
> 0. Since
0 < a < 1 we know that 1/a > 1, so that by the
previous proof we can find an N with
But then it follows that
-
an < for all n > N
This proves that the sequence {an} converges to zero.
- Case -1 < a < 0:
- By the above proof we know that
| an | converges to zero. But since
-| an | < an < | an |,
the sequence {an} again
converges to zero.
- Case a < -1:
- Extract the subsequence {a2m} from the
sequence {an}. Then this sequence diverges to infinity by
the first part of the proof, and therefore the original sequence can
not converge either.
- Case a = 1:
- This is the constant sequence, so it converges.
- Case a = -1:
- We have already proved that the sequence
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does not converge.
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