Example of Proof by Induction 3: n! less than n^n

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Precalculus: Using proof by induction, show that n! is less than n^n for n greater than 1. We use the binomial theorem in the proof. Also included is a direct proof.
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Thank you so much for including the direct proof. Exactly what I was looking for.

proxy
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Thank you sir! This video is really helpful because I have the same question in my book and I don't know how to solve it! Now I know! Thank you so muuuch!

bounieee
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I love that you explained it step by step, very nice pace and very easy to follow. Thank you so much! x

DO-rshq
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@alullabyofpain Yes, ln(x) is logarithm with base e. Sometimes it is written log(x), which is more typical for base 10.

e arises in application like population growth, radioactive decay, and compound interest. The functions for these look like f(x) = C r^x. e appears when we try to differentiate.

Two good calculator exercises: lim_{n-> infinity} (1+1/n)^n = e, so check with large n, and
1 + 1/1! + 1/2! + 1/3! + 1/4! + ... = e, so add the first few terms to check. - Bob

MathDoctorBob
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You're welcome! Glad to be of help.

MathDoctorBob
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I have a whole arsenal of inequality tricks if I ever get to real analysis.

MathDoctorBob
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I think set theory is a good insight into inequality logic: a simple example would be to show how the set of n > 100 is a subset of n > 10 over lN (actually this has application in Math Induction. For example one can immediately use "If k > 5, then 5k > 5" in those kinds of proofs. And I have, as well considered building from the basic axioms of inequalities over lR and definitions of course already taking for granted basic definitions such as 5 > 1 and 1 < 5 and 0 = 0.

chordsequencer
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@MathDoctorBob the function diverges (from 2), but if we wanna find the point on the infinite, by induction i know the larger the n the smaller the number on 1/n so 1/n tends to 0, since then i have (1 + 0 )^infinite, which this function is a constant so the function on the infinite is 1, but I dont find in the behavior of the function the converge to 1 actually i find to diverge, tending to grow larger since we have the same growth rate on n exponential grows faster than the division on 1/n

alullabyofpain
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Thanks for the video, it really helped me out!

Plepple
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@peturie Are you asking for the eskrima stick back? - Bob

MathDoctorBob
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professor i dont think it has anything with this, but I've been wondering, the ln is log on base e right? i heard that e is a constant like pi, where does it comes from? like pi comes from the division by the `round` of the circle divided by the diameter (my english is indeed bad haha) thxx professor! keep up the good work

alullabyofpain
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mathematical induction to prove (n!)^2》n^n

erkanakbas
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what if you consider n! <= n^n
if n=k+1?

elishasli
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CUANTO TIEMPO ESTARE BANEAOOO CALCULALO - ME CABREO MUCHO Y ABANDONO Y SOY MU MALO CALCULA

YKoo
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Hi Bob
I can’t find your first 2 videos on proof by induction???

moragclark
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I feel as though much inequality theory needs to show up here...

chordsequencer
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MAESTRO ME PERDI EN LA PALABRA TRUSCONIDARXONIACONOXIA

YKoo
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11n+2 + 122n+1 is divisible by 133 plz yai data do

kamleshkb