today we are going to do square root of 2 factorial and of course the usual definition n
times n minus 1 times data times 2 times 1 that's only good for plus the whole number so that's not
going to work for us in order to talk about this we have to extend the concept of factorial by
using a very nice integral and let me write that down right here for you guys so let me put down
x factorial and this right here is equal to the integral from 0 to infinity t to the x power and
you multiply this by e to the negative t power d t and this integral is from the pi function with the
gamma function and yes i have a lot of videos on those things already and with this right here you
can do a lot more with factorials such as we can talk about one half factorial you can talk about
negative one half factorial you can talk about pi factorial you can talk about e factorial
of course we can also talk about square root of two factorial but you also to be careful sometimes
we cannot do certain factorial such as negative one factorial that actually doesn't have an answer
but once again i'm going to leave all that to you guys today this is the do so ladies
and gentlemen let's welcome square root of 2 factorial this right here is nicely equal to
the integral from 0 to infinity t to the square root of the 2 power times e to the negative t
dt and yes this right here is an improper integral because we go from zero to infinity
but this right here nicely converges so we have a finite value for that but now
i don't think i can actually do integration by parts whatever integration technique to actually
integrate this and then have an answer in the close form i don't know how to do that if you
guys know how to do it leave a comment down below and let me know but i do have a very nice question
for you guys also continue watching don't just go to the wolfram alpha right away all right
so let me just pause it right here first of all we will be doing an approximation for this and
this is improper integral so to approximate an improper integral of course we will have to use a
proper integral to do so and the trouble of this right here is because of infinity so
this is going to be my first try and let me just put this down right here for you guys
to approximate this let's talk about the integral from 0 to just 10 right let me
just pick this for you guys and of course the inside stays the same we have all this and now
this is a much better integral to deal with we can do some approximation and perhaps
the best approximation technique we have talked about so far is the simpson's rule so we'll
do that right here so suppose we are going to use simpson's rule by simpsons
and when you use simpson's rule you have to pick an n value and let's just say n is equal to 20.
and i'm not going to write down the answer right here for you guys and you might be
wondering this is improper integral going from 0 to infinity and i just use from 0 to 10.
is that even enough even though i say n is equal to 20. well i have another
proposal for you guys this is going to be my second try suppose i go from 0 to 20
and of course the inside stays the same so this time you see i have a longer interval
so of course this perhaps is better compared to that right because this is going from 0 to 20
this is only going from 0 to 10. we need to go from 0 to infinity right but anyway the
trade-off that i will make right here is that i will still use the simpson's rule just to be fair
but this time i would just pick n to be 10 okay so my question for you guys is that
which one right here do you think it's actually a better approximation to this leave a comment
down below and let me know you see the first one or the second one go ahead comment down below
all right i'll trust you guys okay cool
so you guys can find more links in the description because i have videos on the simpsons rule already
and i will just tell you guys what the answers are all right if you go from 0 to
10 with n equals 20 the answer that you get is approximately 1.2560 and so on of course
and if you go from 0 to 20 which is the longer interval and
if you just use at least equal to 10 the answer is approximately 1.2305 and so on
well of course you wouldn't know which one is better unless we have the actual answer
right here right and of course we will have to use one alpha for this so i will just put this
down by wall from alpha the answer that you get on this is approximately 1.2538
so as you can see the battle of choice is actually this one right here this one
wings even though you have a smaller interval but if you have a bigger n value maybe it's better
than having a longer interval with less m value right and perhaps the moral rule behind this story
is that sometimes you may just want to focus on certain things and then just do really well
on them instead of trying to do lots of things and then just do like half jobs and all that
here's a quick reason why from 0 to 10 is actually just enough well when you have e to the negative t
this right here multiply with this in fact this actually converges pretty quickly so from 0 to
10 is pretty good and once again if you focus on this interval with a bigger m value
yeah you'll get a better answer okay thank you so much for watching and if you like to
have more questions to work with then i would highly recommend you to check
out brilliant.org especially now they have this new daily problems and let me show you
guys this one right here is math and logic and this one is in probability here's the question
we're trying to get one two three four five six on the dice and which one is more likely should
we just use one die and roll six times or should we get six dice and roll them at the same time
of course you guys can choose the answer here and i will leave this to you guys and if you guys like
this question you guys can also go to left because you guys can click on the quiz and you can see
more similar questions and if you guys would like to sign up for their courses which are decided in
an interesting and productive way you guys can use the link blenderworks blackberry pen because
this way you guys can get 20 of discount and you can get access to all of these courses which are
really really cool keep in mind practice is the key to get better and practice right it's also a
crucial part use blend down work so you will enjoy the learning process and once again thank you
guys so much for watching the video and as always that's it i will do the eye factorial for you guys
soon