Limit of infinite function

[old hamlet]

The task is to find the limiting value of sqrt(1+sqrt(1+... Infinitely

[civic glacier]

Well we can substitute sqrt(1+sq(1+sq(1+…) for r to get

r = sqrt(1 + r)

[old hamlet]

yes, and then?

[civic glacier]

Solve for r

[old hamlet]

thats all?

[civic glacier]

Yea pretty much

[old hamlet]

well that gives u the golden ratio

[civic glacier]

Well then that’s your answer

[old hamlet]

thanks that solves that part

sadly tho it says it wants you to rigorously find an exact value for r by constructing an appropriate series (a_n) n of the natural numbers and prooving its convergence

how do I do that?

[civic glacier]

Set a_0 = 1, and a_n = sqrt(1+a_(n-1)) for n >= 1

sqrt(1+x) > sqrt(x), so a_n > a_(n-1)
sqrt(1+x) - sqrt(x) decreases as x -> inf, so a_n - a_(n-1) decreases as a_n increases
Therefore a_n - a_(n-1) decreases as n -> inf

sqrt(1+x) - sqrt(x) approaches zero as x -> inf , so a_n = a_(n-1) as n -> inf, therefore at inf, a_n = a_n-1

r = a_inf, so r = sqrt(1+r)

Solve r to get r

[old hamlet]

thanks sm thats really helpfull

could I send u a picture of it once I have it written down? just to make sure I get all subscripts in the right place

or in here idm

[civic glacier]

Sure, just send it to me here

You can ping me once you send it aswell

[old hamlet]

sure, will do

thanks again

[old hamlet]

@civic glacier

[civic glacier]

I don’t understand it but from what I can understand it seems right

[old hamlet]

yeah dw about the german its correctly translated