# Video: Find a Coefficient in a Taylor Series Using a Known Taylor Series

The Taylor series for cos π₯ centered at π₯ = 0 begins as follows: 1 β (π₯Β²/2!) + (π₯β΄/4!) β (π₯βΆ/6!) + .... What is the coefficient of π₯Β² in the Taylor series for π₯Β² cos π₯Β² centered at π₯ = 0?

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### Video Transcript

The Taylor series for cos of π₯ centered at π₯ equals zero begins as follows. One minus π₯ squared over two factorial add π₯ to the fourth power over four factorial minus π₯ to the sixth power over six factorial and so on. What is the coefficient of π₯ squared in the Taylor series for π₯ squared cos of π₯ squared centered at π₯ equals zero?

So we need to find the Taylor series for this function, π₯ squared cos of π₯ squared. We of course know the general form of a Taylor Series where we can find the necessary derivatives and substitute them into this general form. However, weβve been given the series for cos of π₯ centered at π₯ equals zero in the question. So, actually, we can use this instead. Thereβs a really useful technique to find the Taylor series of harder functions by using the Taylor series of common functions just like cos of π₯.

If we take this Taylor series and replace π₯ with π₯ squared, weβve got the Taylor series for cos of π₯ squared centered at π₯ equals zero. Itβs really useful when weβre doing this to keep whatever weβre substituting in brackets. So letβs simplify these exponents first. We remember the index law that tells us if we raise something to a power and then raise that to another power, we can just multiply the powers together. So thatβs the Taylor series for cos of π₯ squared. But remember, weβre really looking for the Taylor series of π₯ squared multiplied by cos of π₯ squared.

So we can do this by multiplying each term by π₯ squared. This is because weβre multiplying the whole series by π₯ squared to get the Taylor series for π₯ squared cos of π₯ squared centered at π₯ equals zero. We can do this by multiplying each term individually by π₯ squared. We can then simplify this. Remembering that if we have π₯ to the π power and we multiply it by π₯ to the π power, this is the same as π₯ to the π add π power.

Remember that the question asked us what is the coefficient of π₯ squared. So we can see here that π₯ squared is on its own. So we can say that the coefficient of π₯ squared in the Taylor series for π₯ squared cos of π₯ squared is one.