Radius of Convergence: The Exponential Function

[Music] good morning fellow mathematicians welcome back to now this video i'm going to censor that because good news we are keeping things family friendly now just because i want a youtube algorithm to favor my channel once again because you see no one's watching my stuff anymore i've had this problem for quite a few months now but it's even worse now only like 1 500 people are watching my videos now every video so that's quite alarming so never mind that was just a joke by the way so not family friendly stuff we are going to do taylor series stuff once again today and we are going to calculate our first radius of convergence today for the exponential function what's the radius of convergence well if you plug a little value into your taylor series for example in your series expansion we don't want the series expansion to explode to infinity otherwise it would just diverge so we want to find out all the values which are the radius of convergence where our well taylor series would converge for this we are going to make use of one theorem out of many you can use koshi hadamar or many different ways to do so but we are just using the well ratio test you have learned in your cake one class for example so let's take a look at the exponential function at first so x of i don't know lambda this time is nothing but the infinite series expansion of lambda to the nth power over n factorial and you see a series is nothing but the infinite summation of many many sequence members where those are our sequences a n for example and the ratio test is as follows we are going to take the limit as n approaches infinity of the absolute value of well a n plus 1 over a n that's why it's called the ratio test and if this ratio is strictly less than one well then this thing converges absolutely at that so we want this condition to hold and why not plug all the sequences into here so for a n plus one we are just going to plug in n plus one to here and for a n we are just going to use well the sequence right here you see so then we have the limit as n approaches infinity of where we have the absolute value of lambda to the n plus one power over n plus one factorial over and now we have lambda to the nth power over n factorial so this looks quite like a mess but everything is going to cancel out quite nicely so this and that is going to cancel out to lambda to the first power so that's the limit as n approaches infinity of the absolute value of just lambda to the first power this is just lambda and now we would have n factorial that's complex fraction over n plus one factorial but n plus one factorial is nothing but n plus one times n factorial so the n factorials little reference are going to cancel out to just well over n right here you see ns element of the natural numbers strictly element of natural numbers so the absolute value of a natural number is just the natural number itself so let's get rid of this right here and now what is going to happen if n approaches infinity well this fraction is going to get smaller and smaller so this overall it doesn't matter which value for lambda you plug in a complex value a real value it really doesn't quite matter this fraction is always going to go to zero in the limit when n approaches infinity [Music] and well that means our radius of convergence so all the absolute values of lambda is just infinity so it really doesn't matter what value you plug in it's always going to converge for every value and that's it so the radius of convergence is infinity in this case it doesn't matter what you plug in yeah and that's basically it like i said there are many more fear rims out there you could make use of but that's the easy choice i guess for most taylor series expansions if you enjoyed this video please like and subscribe recommend the channel if you like make sure to activate the bell button and to share those videos everywhere it really helps the channel out i just want my videos to get a bit more recognition because it's really annoying i have 30k subs now and yeah only like five percent of the people are watching my stuff now it's really quite weird maybe it's just not interesting enough i don't know like hell i care i i do care i wouldn't talk about it if i wouldn't care but well i just like making videos for you guys and yeah well that's it then i guess fingers watching up until the next video have a flammable day i guess ciao can do the foreign [Music]

Channel: Flammable Maths

Views: 8,192

Keywords: flammable, maths, integration, hard, university, potsdam, papa, flammy, putnam, integral, Maclaurin, Expansion

Id: jWGb3oDRc3Y

Length: 5min 27sec (327 seconds)

Published: Tue Nov 27 2018