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Current time:0:00Total duration:4:25

Converting explicit series terms to summation notation (n ≥ 2)

Video transcript

let's say that we're told this some right over here where our index starts at two and we go all the way to infinity that this infinite series is negative 8/5 plus sixteen over seven minus thirty-two over nine plus and we just keep going on and on forever and so what I want to do is find is to explicitly define what a sub n here is here so right now we just say if you take the sub of a the sum of a sub n from N equals two to infinity it turns out you get this sum right over here but let's think about what a sub n how we can actually define it in terms of N and I encourage you to pause the video right now and try it on your own so the first thing that you might realize is well this is the number that we're going to get this is the number that we're going to get let me write it this way a sub 2 a sub 2 is equal to negative 8/5 a sub 3 a sub 3 is equal to 16 is equal to 16 over 7 a sub 4 is equal to negative 32 negative 32 over 9 and I'm just I'm just giving the sign to the number in the numerator negative 8/5 is the same thing as negative 8 over 5 let me make that a little bit clearer so make that a little bit clearer so this is negative 8 over 5 obviously this is positive so I don't have to really worry about too much and then here I'm just saying negative 32 over 9 to the same thing as negative 32 over over 9 so let's see let's see if we can first find a pattern in the numerator a pattern in the numerator so when we go from negative 8 to 16 what's happening well we're multiplying we're multiplying by negative 2 we're multiplying by negative 2 now to go from 16 to negative 32 we're multiplying by negative 2 multiplying by negative 2 again so you might say okay well whatever we have in the numerator must be a power of negative 2 all right if you say well maybe this is negative 2 squared well you know that negative 8 isn't negative 2 squared negative 2 squared is equal to positive 4 negative 8 this right over here negative 8 that is equal to negative 2 to the third power 16 is equal to negative two to the fourth power negative 32 is equal to negative negative two to the fifth power so notice our exponent on the negative two is always going to be one more one more than our index our Nexus to our exponent is 3 our index is 3 our exponent is 4 and Nexus 4 our exponent is 5 so that gives a sense at least the numerator the numerator is going to be whatever our index is is going to be so let me write this down so a sub n a sub n is equal to well it's going to be negative 2 to whatever index we're at to that index plus 1 power so that's one way to think or that's a reasonable way to think about our numerator now let's think about our denominators so over so we go from 5 so when n is 2 we're at 5 when an N is 3 where it's 7 when n is 4 we're at 9 so notice 5 is 2 times 2 plus 1 is 2 times 2 2 times 2 plus 1 this right over here is 2 times 3 plus 1 this right over here is 2 times 4 plus 1 and you should just kind of play around with different patterns in your head until you say hey well look this is let's you know this is increasing by 2 every time notice this increases by a by 2 every time but these aren't exactly these aren't exactly multiples with 2 these seem to be off by off of 1 more than the multiples of 2 which is a good sign that this is going to be 2 times our index plus 1 so we could write this down we could write this down as 2 times our index plus 1 and we're done that's what a sub n is and if we wanted to write this series in Sigma notation we would write this as a sum from N equals 2 to infinity of negative 2 to the n plus 1 power over 2 and plus 1 and that would equal this series right over here