Methods from Sequences and limits
All methods from Sequences and limits.
Simplify a sequence limit
For a sequence limit:
- simplify algebraically, dividing by the dominant power of when appropriate;
- rationalise when a square root creates a difference of large, nearly equal quantities;
- rewrite positive powers using logarithms and exponentials;
- use the squeeze theorem when direct limit laws do not apply;
- for a recursively defined sequence, prove convergence before solving the equation obtained by taking its limit.
The order matters for recursive sequences. Solving a fixed-point equation first only finds possible limits; it does not prove that the sequence converges to one of them.
Analyse a recursive sequence
For a recursive sequence:
- find candidate limits by solving the fixed-point equation;
- prove a bound that is preserved by the recursion;
- prove monotonicity on that bounded region;
- invoke the Monotone Convergence Theorem;
- use continuity to substitute the now-established limit into the recursion;
- select the algebraic root compatible with the bound.