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Showing Recursive Sequences Converge

Introduction

Many students have asked me about how to prove that a recursively defined sequence

converges. Hopefully, after reading these notes, you will be able to tackle any such

problem.

Preliminaries

To prove that the recursive sequence { }na converges, we need to show that it is bounded

and monotonic. In practice, we either show thatn

a is increasing and bounded above, or

decreasing and bounded below. For now, lets assume that we are trying to show that our

sequence is increasing and bounded above.

{ }na is bounded above means that there is a numberMsuch that na M< for all n.

Since there are infinitely many different values ofn, proving any statement containing the

phrase for all n is going to be difficult. Our strategy is to use mathematical inductionto prove this statement. We will then use our bound to show that the sequence is

increasing.

7/30/2019 Showing Recursive Sequences Converge

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Mathematical Induction

Imagine a ladder with infinitely many rungs. If I can show you how to get onto the

bottom rung of the ladder and show you a general procedure for getting from any rung tothe next one, then you can get to any level on the ladder. Heres how to get to rung 3.

We can get to rung 1 because we know how to get on the ladder.

We know the general procedure for advancing one rung; so we can get from rung 1 to

rung 2.

We can also use that general procedure for getting to rung 3.

Using the same argument, we can demonstrate how to get to rung 4, rung 5 and even rung657. In fact, for any n, we can get to rung number n.

We have just demonstrated that for all n, we can get to rung n.

Our strategy for sequences will be similar. For example, if we want to show that the

sequence { }na is bounded above by 2, we show

1 2a <

1 2

2 3

3 4

2 2

2 2

2 2

...

a a

a a

a a