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# Worked example: cosine function from power series

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.E (LO)
,
LIM‑8.E.1 (EK)
,
LIM‑8.F (LO)
,
LIM‑8.F.2 (EK)

## Video transcript

so I have an infinite series here the sum from N equals 0 to infinity of negative 1 to the N power times X to the 6 and over 2 and the whole to n factorial and my goal in this video is to evaluate this this power series when X is equal to the cube root of PI over 2 and I encourage you to pause this video and give it a go on your own and I will give you a hint the key to this is to figure out well what function is this the power series for and then use that function to evaluate this and there's another clue here is that they this this is kind of a mysterious or a suspicious looking number here PI over 2 that looks like something I would use a a trig function to evaluate it that might be a little bit more straightforward so I'll let you have a go at it so I'm assuming you have tried so let's try to work through this together and any of these types of problems I like to at least expand out this power series so I get a better sense of what it what it's like so this right over here if I were to expand it out this is going to be equal to when n is 0 this is 1 actually does all of these are 1 so it's just going to be 1 when n is 1 it's going to be negative 1 X to the sixth X to the sixth over 2 factorial when n is 2 it's going to be positive negative 1 squared is positive 1 times X to the 12th over 4 factorial and then let's just do one more when x is equal to 3 it's going to be negative x to the 18th X to the 18th over 6 or 6 factorial and you just keep going on and on forever now offhand I don't know a function especially a trigonometric function because that was kind of our clue here this pi over 2 makes me feel like I might this might be a trigonometric function right over here nothing jumps out at me offhand but but this does look suspiciously familiar this looks awfully Co close to the power series or the Maclaurin series for cosine of X which we have seen multiple times let's just remind ourselves what that is and if this doesn't look familiar there the previous video where I do the Maclaurin series for cosine of X goes into detail on how I get this the Maclaurin series for cosine of X is equal to so I'll just write a few terms so I'll write approximately equal to 1 minus x squared over 2 factorial plus X to the fourth plus X to the fourth over 4 factorial minus X to the sixth over 6 factorial and just like that you're probably seeing the similarities well the first term is the same the sign negative positive negative positive negative positive negative positive 2 factorial 4 factorial 6 factorial the difference is the the powers the exponents on the axis this is x squared this is X to the 6 this is X to the fourth that's X to the 12 this is X to the sixth that's X to the 18th well what if we what okay so is some something for you to think about is well how can we can we replace X with something here because anything that I change if I take cosine of but if I change X to R I know a plus B everywhere we see an X you would replace it with an A plus B can we put a power of X here so that these things end up like that well this X is 6 is the same thing X to the sixth is the same thing as X to the 3rd squared that's X to the 3rd squared this right over here X to the 12th is the same thing as X to the 3rd to the 4th power this right over here is the same thing as X to the 3rd to the 6th power so if we could replace each of these X's with X to the 3rd we will get this power series up here well how do we do that well we would just say well it's a cosine of X to the 3rd and actually let me do that in a different color so the cosine and that's not a different color so a cosine of X to the third is going to be equal to and once again every see where we see an X we replace it with X to the third so it's one minus and actually I'm just going to put a parenthesis squared two factorial which I wanted to do that in the green let me do all of this in the green all right so it's going to be equal to one minus parenthesis squared over 2 factorial plus parenthesis to the fourth power over 4 factorial minus parenthesis to the sixth power over six factorial and now let me get change back to that move color and since I'm doing the cosine of X to the third well this is going to be X to the third squared this is going to be X to the third to the fourth power this is going to be X to the third to the sixth power which is exactly what I have right over here so this right over here is the power series for cosine of X to the third so evaluating this when X is equal to the cube root of PI over 2 is the same thing as evaluating this when x is equal to the cube root of pi over 2 let me write that down because this is interesting so this so I'll just rewrite it from N equals 0 to infinity of negative 1 to the N X to the sixth n over 2n factorial this is equal to this is the power series representation of cosine of X to the third power so if you want to evaluate this when X is a cube root of PI over 2 we just have to evaluate this when X is a cube root of PI over 2 and this does suspiciously work out nicely because if you take the cube of the cube root well good things happen so the cosine of the cube root of PI over 2 to the third power well that's just the same thing as cosine of PI over 2 which of course is equal to is equal to 0 and we are done