example of converging increasing sequence
Since , the two first above equations imply that . By induction on one can show that
The numbers are all below a finite bound . For demonstrating this, we write the inequality in the form , which implies , i.e.
for all . We study the polynomial
From its latter form we see that the function attains negative values when and that increases monotonically and boundlessly when increases from 1 to . Because as a polynomial function is also continuous, we infer that the equation
has exactly one root (http://planetmath.org/Equation) , and that is negative for and positive for . Thus we can conclude by (2) that for all values of .
The proven facts
settle, by the theorem of the parent entry (http://planetmath.org/NondecreasingSequenceWithUpperBound), that the sequence
converges to a limit .
Taking limits of both sides of (1) we see that , i.e. , which means that , in other words: the limit of the sequence is the only of the equation (3).
- 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
|Title||example of converging increasing sequence|
|Date of creation||2013-03-22 17:40:44|
|Last modified on||2013-03-22 17:40:44|
|Last modified by||pahio (2872)|