Example:
The infinite series
.gif)
=
1/2 + 1/4 + 1/8 + 1/16 + ... converges to 1 (this series
is a special case of the geometric series).
The n-th partial sum for this series is defined as
- S n =
1/2 + 1/2 2 + 1/2 3 + ... + 1/2 n
We need to find a closed form for this expression to be able to
take the limit of the sequence of partial sums.
If we divide the above expression by 2 and then subtract it
from the orginal one we get:
- S n - 1/2 S n =
1/2 - 1/2 n+1
Hence, solving this for S n we obtain
This is now a sequence, and we can take the limit as n
goes to infinity. By our result on the power sequence, the
term 1/2 n+1 goes to zero, so that
That proves, by definition, that the infinite series converges
to the number 1.
To Theory |
Glossary |
Map
(bgw)