# Proving limits of recursive sequences using definition

• ricardianequiva
In summary: If it's so obvious, then prove it.There are generally two steps in these kinds of problem : 1° Suppose the limit exists and find its value (or possible values). This usually involves noticing that if a_n-->a, then a_{n+1}-->a and solve for a algebraically in the equation you get by taking the limit of both side in the recurrence relation.. for instance here we get a = a^(2/5).2° Prove that the limit exists, and in the event that step 1 lead to multiples possible values for the limit, determine which one is right.
ricardianequiva
in general, how does one go about proving limits of recursive sequences using the definition?

for example, how does one prove a_n+1 = (a_n)^2/5 => lim(a_n)=0?

For me it's obvious but the TA insists that things like that require proving?

ricardianequiva said:
For me it's obvious but the TA insists that things like that require proving?
If it's so obvious, then prove it.

There are generally two steps in these kinds of problem :

1° Suppose the limit exists and find its value (or possible values). This usually involve noticing that if a_n-->a, then a_{n+1}-->a and solve for a algebraically in the equation you get by taking the limit of both side in the recurrence relation.. for instance here we get a = a^(2/5).

2° Prove that the limit exists, and in the event that step 1 lead to multiples possible values for the limit, determine which one is right.

ricardianequiva said:
in general, how does one go about proving limits of recursive sequences using the definition?

for example, how does one prove a_n+1 = (a_n)^2/5 => lim(a_n)=0?

For me it's obvious but the TA insists that things like that require proving?

Why, would that be obvious! Unless you are some kind of genius, so you can see all the steps to the end in your head. Otherwise that is not obvious unless you start doing it!

recurses!

ricardianequiva said:
for example, how does one prove $$a_{n+1} = (a_n)^{2/5}$$ => lim(a_n)=0?

Hi ricardianequiva!

In a simple case like this, where you have a recursive definition, just "recurse" it!

In this case, $$a_{n+k} = (a_n)^{{2/5)^k}$$.

Does that do it for you?

ooh! … just noticed … it wasn't obvious that lim a_n = 0 … in fact, it wasn't even true! … i think there's a moral there, somewhere!

Last edited:
ricardianequiva said:
in general, how does one go about proving limits of recursive sequences using the definition?

for example, how does one prove a_n+1 = (a_n)^2/5 => lim(a_n)=0?

For me it's obvious but the TA insists that things like that require proving?
Good! You have a good TA.

Actually, as Tiny Tim pointed out, this is not only not "obvious", it's not even true. What this sequence converges to, and even whether or not it converges, depends on the initial value and you did not give that!

## What is a recursive sequence?

A recursive sequence is a sequence of numbers where each term is defined in terms of the previous terms. The first term, or initial value, is given and subsequent terms are calculated using a rule or formula.

## How do you prove the limit of a recursive sequence using definition?

To prove the limit of a recursive sequence using definition, you must show that as n approaches infinity, the terms of the sequence get closer and closer to a specific value, known as the limit. This can be done by showing that the difference between consecutive terms approaches 0 as n approaches infinity.

## What is the definition of a limit for a recursive sequence?

The formal definition of a limit for a recursive sequence is: For a given sequence {an}, the limit L is said to exist if for any positive number ε, there exists a natural number N such that for all n ≥ N, the absolute value of an - L is less than ε.

## What are some common examples of recursive sequences?

Some common examples of recursive sequences include the Fibonacci sequence, the factorial sequence, and the geometric sequence.

## Why is it important to prove the limit of a recursive sequence?

Proving the limit of a recursive sequence is important because it helps us understand the behavior of the sequence as n approaches infinity. It also allows us to make predictions about the long-term behavior of the sequence and can be used to solve real-world problems in fields such as economics, physics, and biology.

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