Hello, everybody. This is Kevin Truck with new Murad. Let's look at what this infinite Siri's might add up to. Well, there's a lot that's going on for this, but we do know that we're gonna be starting off with is is putting in a value of one and then indexing it every single term from there. So we've got one times one plus one. It's to over one plus two is three plus one times one plus three would be four. We're going to get something along. These lines will be adding that to the next index well behind right that we have two times three and then each of this is going to go up by one is also four times five, and we're gonna continue to add that through over and over and over again. Now, notice that we do have what looks like a denominator that is gonna be larger than the numerator. But instead of trying Teoh solve all this out, we can do what's called an end value on term test to find out whether or not this is going to decrease fast enough that I don't accidentally blow up to infinity. And so here's kind of what that looks like. Let's consider if at the end of all of this, at the end of all of this, I am going to be still adding terms that are not zero village, which there is no end. So really, what? We're talking about its limits here. So let's structure and thought little bit more properly. Let's look at what is the limit as n goes to infinity of end times and plus one over and plus two times and plus three. And really, what we're checking by doing this is we're saying, If I continue doing this forever, never, never, never will I still be adding things. Or will I theoretically eventually be adding zero at my forever term? So let's go ahead and work this on out. What? We can recognize that if I was just to kind of plug this and I'm gonna write this in brackets, because this is this is me being sloppy, but it's a nice little note to give yourself. I plugged in infinity here, which you can't do. It's not a number, but if we're going Teoh allow ourselves a little bit of error, we would see that we're gonna have infinity over infinity again. Not proper mathematics here, but we can recognize that we do have it's called indeterminate form. And so we might use what's called Loba tells rule to simplify out this This problem so low Beatles Rule says that if I have something which would have evaluated out to infinity over infinity, if I was incredibly sloppy, that I can simply do the derivative of the numerator and then the denominator. And that will be a much easier, much easier ratio to consider as I do the limits. Okay, so we're gonna do a product rule in top because we do see that we have end times n plus one. It's gonna be derivative of the first times. The second as is so it's gonna be endless one plus the first, as is times the driver of the second, which is gonna be one over. And then the bottom is going to the same thing driven or the first is gonna just simply be one times the second, as is, plus the derivative of the first times. The second as is now, I did forget some notation here. Remember that this Onley true love hotels rule on Lee works If you are still talking about limits, you're not allowed to step out of limit thought here. So all this is going to simplify out to sea Teoh. Let's see n plus one plus ends, too, and plus one over and then we have en plus two plus n plus three. So to end, plus five now we could actually do Loki tells Rule one more time. Gonna be a little bit simpler this time. Notice that there is a way to evaluate this by simply dividing by the largest and term over here. But I'll leave that as a as a exercise for another video. Lovato's real one, more time driven with the numerator is going to simply be too driven other than denominator is simply going to be, too. We're gonna include the fact that we're talking about the limits here, but the limit as N goes to infinity of 2/2 is just gonna be the limit as N goes to infinity of a constant. So it's just going to simply be that constant. So this is going to be ah, value of one, and what we can say is that because if we re contextualized all the math that we've done, the forever term, if you will, the forever term that I am adding into this series is one. So I'm adding a forever number off ones or things that are not zero is a very loose way of interpreting it. But essentially, what we can conclude is that because this serious does not decrease down to zero ever that I am going Teoh diverge. And this entire Syria's is going Teoh blow up to infinity. So we can kind of just put a little notation here and say that this thing, if I was gonna continue it forever, would approach infinity and grow in value. Okay, it's going Teoh diverge. So you could say maybe something like diverge to or towards infinity.