2015 AP Calculus BC 6b | AP Calculus BC solved exams | AP Calculus BC | Khan Academy

Part B write the first four nonzero terms of the mclen series for f prime, the derivative of f. Express f prime as a rational function for the absolute value of x being less than R, our radius of convergence.

So if we want to find f prime, we could just take each of these terms with respect to x. We could just say if this is f, then f prime... and I'm not going to scroll down just so I can see this up here. We could just say f prime of x is going to be…or the mclen series for f prime of x.

Maybe I should write it that way, so let me write it this way: mclen series for f prime of x. Well, it's going to be the sum from n = 1 to infinity, and we would just take the derivative of this right over here with respect to x.

And so this is just an application of the power rule. Take the exponent right over here, multiply it by the coefficient. So if you take n times this, it cancels out with this n, so it's going to be -3 to the n minus 1, and then decrement your exponent times x to the n minus 1.

And so they want us to write the first four nonzero terms of the mclen series. So that is going to be equal to…so I'll write approximately equal to because we're only going to write the first four terms of this infinite series.

And just to be clear what I did, I just did the power rule here. I looked at this exponent which is n multiplied by this coefficient which had an n in the denominator. So that n and this n cancel out, so I'm just left with -3 to the n minus 1, and then I decrement that exponent. That's straight out of the power rule, one of the first things you learn about taking derivatives.

And so if we want the first four non-zero terms, when n equals 1, this is going to be -3 x to the 0 power. Let me just write it, -3 to the 1 minus 1 times x to the 1 minus 1. That's when n = 1 plus -3 to the 2 minus 1 times x to the 2 minus 1.

And actually, I could have just written this as -3 x to the n minus 1. Actually, let me do that just for so this is…I could just write because they have the same exponent. This will simplify a little bit as -3 x to the n minus 1.

And so this is going to be approximately, when n is equal to 1, this is going to be equal to zero. So -3x to the zero power is just going to be one. When n is equal to 2, this is going to be 2 minus 1, so it's going to be the first power, so -3 x to the first power.

So I could just write this as -3 x, and then when n is 3, well this is going to be -3 x squared, so -3 x squared is going to be -3 times 2 is 9 x squared.

And then the fourth term is going to be -3 x to the 4 - 1 power, so to the third power. So 3 to the 3 power is -27 times x to the 3 power. So there you have it, that's the first four nonzero terms of the mclen series.

You could have also just looked over here and said, okay the derivative of x with respect to x is 1, the derivative of -3 x squared with respect to x is -3 times 2, so you could have said the derivative of this is 9 x squared right over there, and then you would have had to write out the fourth term and take out this the derivative in the same way and you would have gotten this right over here.

So we did the first part. We wrote the first four nonzero terms of the mclen series for f prime, the derivative of f. And then they say express f prime as a rational function for the absolute value of x being less than R.

So this sum, if we assume it converges, and we know the radius of convergence already, so assuming that we're dealing with x that are within the radius of convergence, so this right over here you might recognize this.

So I could write it like that, or I could also write it. I could also write it if I take…if I start at n equals 0. So I could also write this as from n equal 0 to infinity of -3 x to the n, because now the first term was to the zeroth power. That first term was to the zeroth power, so whether you do 1 minus 1 is where you start or you just start at zero, these two things are equivalent.

You might recognize these as a geometric series with a common ratio of negative 3x. And so what's the sum of a geometric series with a certain common ratio? Well, it's going to be equal to the first term, and regardless of how you view this, the first term is going to be 1 divided by 1 minus the common ratio.

So our common ratio is -3x. So, 1 minus -3x, well that's just going to be 1 + 3x. If what I just did here looks unfamiliar to you, I encourage you to watch the sum of infinite geometric series.

And not only do we show you this formula and how to apply it, but we show how you can prove this formula. It's actually a pretty fun proof. But anyway, regardless of how you view this mclen series, it is an infinite geometric series.

And this, assuming that our x is in our radius of convergence, this is what our sum is; this is what we are going to converge to.