Convergence and Sum to Infinity
In the first lesson on GPs we considered series with
fixed numbers of terms; Sn
What if the series has an infinite number of terms ie. S∞?
Will the sum of the series move towards a fixed number
==> convergence…
… or will the sum get forever larger
and larger ==> divergence?
What does this depend on?
A converging sum
To share this square cake between 3
people we can split in into quarters an
infinite number of times – allocating 3
of the quarters and dividing the 4th.
Eventually there will be no cake left
and each person will have the same
amount.
Each person gets
¼ + ¼ × ¼ + ¼ × ¼ × ¼ + ...
= ¼ + (¼)2 + (¼)3 + (¼)4...
This is the sum to infinity of a geometric series with: a = ¼ , r = ¼
Recall
= a/(1 - r)
In this case
Sigma notation ( Σ )is a convenient way of
summarising the sum of any GP.
This one notates sum of all powers of a half starting
with the first.
As n becomes larger and larger, Sn ever closer to the fixed
value, 1.
We say that the “limiting value” of Sn as n “tends to
infinity” is 1; and we write:
lim
∞
n
Sn =1
Since this “limiting value” is a finite number, we say that the
series “converges” to 1.
Not all sequences and sums converge.
They may oscillate but converge … e.g. -1/3, 1/9, -1/27 …..
They may diverge e.g. 2, 4, 6, 8, ….
They may be periodic e.g. 0, 1, 0, 1, 0, 1, …
What would these look like?
Oscillating convergence
The Golden Ratio
Fibonacci sequence