5.7 Formula for the Limit
Once we start getting close to the limit, successive terms get closer and closer to each other, and the difference between two consecutive terms becomes so small that it gradually disappears. When we ‘reach the limit’ we could thus say that two consecutive terms are both the same.
In other words, both Un+1 and Un are equal to each other, and so are both equal to the limit which we can call L. So, at the limit, the equation Un+1 = aUn + b can be written instead as L = aL + b. This can be solved for L as follows:
| 2. Reduce the two terms involving L to just one term, by taking out L as a common factor. | L | = | aL + b | 1. Move both terms involving L to one side of the equation. |
| L - aL | = | b | ||
| L(1 - a) | = | b | ||
| L | = | _b_ |
3. Divide both sides by the bracket (1–a) to get L on its own. | |
(1 - a) |
The last line is an easy one to remember.
