Sequences 4 - Example of Monotone Convergence Theorem

Calculus: We apply the Monotone Convergence Theorem to find the limit of the recursive sequence a_0 = 0, a_n = (1+a_{n-1})/(2+a_{n-1}).

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[ghettomist]

Bruv you look like a pro wrestler. It’s like getting a class from Hulk Hogan. Really inspired, Thank You!

[MathDoctorBob]

@thunderlax47 Thanks for the comment. Q1: First, box is less than 1 + box. That should be believable if you try out a few numbers for box (3 lt 4, 4 lt 5, .5 lt 1.5). Since we assume box is positive, so is 1 + box. Divide both sides by 1 + box to get box/(1+box) < 1. Again try it out with some positive numbers for box to verify.

[MathDoctorBob]

@thunderlax47 (cont.) Q2: The hard part is finding the statement that does what we want, which I get entirely by working backwards. First I notice that the first few terms of a_n are less than 1, so I suspect bounded above by 1. If I set a_n less than 1, setting box = 1 + a_{n-1} in the recursive definition gets the statement.

This is standard operating procedure for inequality proofs and induction, which some calculus classes cover. Of course they take much practice.

[MathDoctorBob]

@Thatdarnmonkeylove You're welcome! Glad to be of help! - Bob

[MaryJane-bolj]

dude this is cool man :)
Its like im getting a motivational sports video for my brain...
"what we can do then is divide both sides of this by 1+box" class xd (and ty really helped for my test coming up)

[MathDoctorBob]

You're welcome! This was a fun one. - Bob

[MathDoctorBob]

@officialIvan You're welcome! - Bob

[HidingTutorials]

Yo i could not for the life of me figure out a limit to a sequence where its terms were defined by previous terms and I was looking all over the internet. Finally got the rite answer thanks to you! Fuck YES!!!

[thunderlax]

In step 2, how is box divided by 1 + box less than 1? And where did the 1 come from initially?

[ragamuffin]

Thanks so much for this, clear and concise, really helped.

[GabSmacked]

Doesn't matter how many times I watch this guy...he still scares me

[GorgeousFortress]

Great video! although I would really like to see the induction proof for the increasing part of the series. I find it very hard to consider asumtions true, and they tend to lead me to make more incorrect ausmtions.