NUMBER_TOKEN factorial is NUMBER_TOKEN NUMBER_TOKEN factorial is NUMBER_TOKEN NUMBER_TOKEN factorial is NUMBER_TOKEN so infinity factorial should be infinity that would make a lot of sense and indeed this product treated as an infinite sequence diverges to infinity but you know i would not have made this video if there wasn't something interesting going on let's take the log of both sides and assuming that we can apply properties of logarithms to break up this log of an infinite product into an infinite sum of logarithms this condensed summation notation looks so good there just has to be a function that represents it enter the infamous riemann zeta function now i don't see any logarithms in this sum but we will after we take the derivative with respect to s the derivative of one over n to the s with respect to s is minus one over n to the s times ln of n now substituting s equals zero gives us the negative of our sum in question so we can say that the natural log of infinity factorial is equal to the negative first derivative of the riemann zeta function at zero there's a bit of a caveat here we're using the analytic continuation of the riemann sata function which is an idea i've touched on in a previous video essentially we're extending the domain of the riemann zeta function to substitute in more values that don't normally make sense now assuming that we actually can plug in zero doesn't actually get us any closer to computing this sum enter the darishlay ada function which has a very nice relationship between it and the riemann zeta function now we're talking about derivatives here so on the left let's take the derivative of the dareshla eta function and on the right we'll have to deal with this product rule keeping in mind all of our rules for differentiating exponential functions after that plug in zero but before i do and since you seem to like learning you should check out the sponsor of today's video skillshare skillshare is an online learning community with thousands of inspiring classes it's great whether you want to learn a new skill expand on your passions or you just feel like being creative if you were ever interested in making your own math videos you can learn everything you need from videography to editing to youtube personally i really enjoyed the productivity master class principles and tools to boost your productivity by ali abdull the class talked about a number of things you don't usually hear about what's great is that you can join with no risk since the first NUMBER_TOKEN NUMBER_TOKEN of my subscribers to click the link in the description will get a one month free trial of skillshare so you can start exploring your creativity today there's no ads they're always launching new classes and what i like best is that you have an unlimited learning potential now take its derivative keeping in mind all of our rules for differentiating exponential functions and now we can go ahead and plug in zero which is what we were really interested in and if we start to write out this sum we get a very interesting pattern if we try to manipulate this with properties of logarithms again remember when we're subtracting logarithms we can write it as a single logarithm of the first argument over the second argument and when we're
START_SPONSOR_TOKEN before i do and since you seem to like learning you should check out the sponsor of today's video skillshare skillshare is an online learning community with thousands of inspiring classes it's great whether you want to learn a new skill expand on your passions or you just feel like being creative if you were ever interested in making your own math videos you can learn everything you need from videography to editing to youtube personally i really enjoyed the productivity master class principles and tools to boost your productivity by ali abdull the class talked about a number of things you don't usually hear about what's great is that you can join with no risk since the first NUMBER_TOKEN NUMBER_TOKEN of my subscribers to click the link in the description will get a one month free trial of skillshare so you can start exploring your creativity today there's no ads they're always launching new classes and what i like best is that you have an unlimited learning potential END_SPONSOR_TOKEN