Proving limits of recursive sequences using definition

[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?

 

[rocomath]

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

If it's so obvious, then prove it.

 

[quasar987]

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.

 

[sutupidmath]

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!

 

[tiny-tim]

recurses!

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!

 

[HallsofIvy]

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!