 So the calculus of series is a lot of fun and it's great mathematics But if you're practically minded you might ask is it actually useful for anything? And so the question we might ask can we use the Maclaurin and Taylor series to approximate functions and the answer is We hope so because otherwise we've done a lot of work for nothing useful Now here's a cautionary tale that suggests we should be very careful So we can find the Maclaurin series for 1 over 1 minus x. It's going to be 1 plus x plus x squared plus x cubed and so on So on the left hand side, I have a perfectly nice respectable function over on the right hand side I have a Maclaurin series So I would like the Maclaurin series and the function to have the same values So well, let's let x equal to on the left hand side. I have 1 over 1 minus 2 And on the right hand side if x equals 2 I have 1 plus 2 plus 2 squared plus 2 cubed and so on and we can simplify And I find that negative 1 is equal to 1 plus 2 plus 4 plus 8 plus Wait a minute. I disbelieve that this is the case and in fact this statement is absolute nonsense And the problem is that this series is Divergent the terms do not go to zero and this illustrates an important point Nothing can be learned from a divergent series Since we have to talk about divergence or convergence of a series. Let's go back to this notion of partial sums Suppose the Maclaurin or Taylor series is convergent for some value of x Then what we'd like is for the sum of the infinite series to be the function value And if that's the case the partial sums approximate the sum of the infinite series But the real question then becomes so how accurately does the partial sum approximate the infinite series? and we can split the problem up into two cases one case is easy and the second case is Unfortunately, also easy The easy case is that if we have an alternating and eventually decreasing series So suppose s is an eventually alternating eventually decreasing series So this s without a subscript indicates the sum of the series then the difference between s and the nth partial sum is going to be less than the n plus first term for a sufficiently large n and The way you can think about it is that the partial sum of an alternating decreasing series is Accurate to within the first excluded term So for example, let's say this series from n equals 1 to infinity of minus 1 to n over n squared And we want to find the sum to within one tenth so we might begin by just writing down the first few terms of the series and We see that the series is alternating and decreasing so the alternating series theorem applies and Again, the way to think about this is that the error in using the first n terms of the series is less than the next term So if I stop with the first term our error will be smaller than the next term the absolute value of one quarter Since we wanted to get our series sum to within one tenth. This is too big So we'll include the second term and in that case our error will be smaller than the next term absolute value of one ninth That's still too big So we'll include that term and so our error using these first three terms is going to be smaller than the next term 116th Since we wanted to find the series sum to within one tenth. The first three terms will give us the desired accuracy How about approximating the value of a definite integral say between zero and one of sine x squared to within 0.001 Now sine of x squared is not a function that we have an anti-derivative for at least not one that we can describe using elementary functions But because it's sine we can use the Maclaurin series for a sine We'll replace x with x squared Which gives the Maclaurin series for sine of x squared? Because the Maclaurin series for sine converges for all values of x then we can integrate term-wise And then evaluate to get our definite integral expressed as a series Now if we write down the first few terms of the series we see that it is alternating and eventually decreasing So we can use the remainder theorem for alternating series So remember the remainder theorem for an alternating series is essentially that our series is Accurate to within the first excluded term So let's say we stop at the first term then the error is smaller than the next term And while this is a small error. It's not small enough So let's include the next term if we stop at the second term the error is going to be smaller than the third term and Since we want our error to be less than 0.001 this will be small enough