Convergent and divergent sequences

let's say I've got a sequence starts at one then let's it goes to negative 1/2 then it goes to positive 1/3 then it goes to negative 1/4 then it goes to positive 1/5 and it just keeps going on and on and on like this and we could graph it let me draw our vertical axis so I'll graph this is our y-axis and I'm going to graph y is equal to a sub N and let's make this our this is a horizontal axis where I'm going to plot our ends so this right over here is our ends and this is let's say this is right over here is positive 1 this right over here is negative 1 this would be negative 1/2 this would be positive 1/2 I'm not drawing it I'm not drawing the vertical and horizontal axes at the same scale just so that we can kind of visualize this properly but let's say this is 1 2 3 4 5 and I could keep going on and on and on so we see here that when n is equal to 1 a sub n is equal to 1 so this is right over there so when n is equal to 1 a sub n is equal to 1 so this is y is equal to a sub n now when n is equal to when n is equal to 2 we have a sub n is equal to negative 1/2 when n is equal to 2 a sub n is equal to negative is equal to negative 1/2 when n is equal to 3 a sub n is equal to 1/3 it's equal to 1/3 which is right about right about there when n is equal to 4 a sub n is equal to negative 1/4 which is right about right about there and then when n is equal to 5 a sub n is equal to positive 1/5 which is maybe right over there and we keep going on and on and on so you see the points they kind of jump around but they seem to be getting closer and closer and closer to 0 which a switch would make us ask a very natural question what happens to a sub n as n approaches infinity or another way of saying that is what is the limit let me just color what is the limit of a sub n as n approaches infinity well let's think about if we can if we can define a sub n explicitly explicitly if we can define the sequence explicitly so we can define the sequence as a sub n where n starts at 1 and goes to infinity with with a sub n equaling what is it equal well if we ignore sine for a second it looks like it's just 1 over n but then it's we seem like we oscillate in signs we start with a positive then a negative positive negative so we could multiply this times negative 1 to the let's see if we multiply times negative 1 to the n then this one would be negative and this one would be positive but we don't want it that way we want the first term to be positive so we could say negative 1 to the n plus 1 power and you can verify this works when n is equal to 1 you have 1 times negative 1 squared which is just 1 and they'll work for all the rest we could write this as equaling negative 1 to the n plus 1 power over n and so asking what the limit of a sub n as n approaches infinity is is equivalent to asking what is the limit of negative 1 to the n plus 1 power over n as n approaches infinity is going to be equal to remember a sub n this is just a function of n it's a function where we're limited right over here to positive integers as our domain but this is still just a limit as something approaches infinity and I haven't rigorously defined it yet but you can think you can conceptualize what's going on here as n approaches infinity the numerator is going to oscillate between positive and negative 1 but this denominator is going to get bigger and bigger and bigger and bigger so we're gonna get really really really really small numbers and so this thing right over here is going to is going to approach 0 now I have not proved this to it yet I am just claiming that this is true but the if this is true if this true so let me write this down if true if true if the limit of a sub n is n approaches infinity is 0 then we say then we can say that a sub n converges converges to zero that's another way of saying this right over here if it didn't if it just if if the limit as n approaches infinity didn't go to some value right here and I haven't rigorously defined how we define a limit but if this was not true if we could not set some limit what doesn't have to be equal to zero as long as if this was not equal to some number then we would say that a sub n diverges