Maclaurin Series and Limits

Nota:

• Maclaurin Series and limits can be found in Chapter 1 of the AQA FP3 textbook
• The two topics often appear in questions together
• There are two key expansions to learn. Basic expansions and the formula are in the formula book.

1. The Theory

   Nota:

   • Maclaurin's Theorum is a very useful way of expressing functions that are otherwise impossible to use calculus on. They also help when calculating limits when used in conjunction with methods from FP1.
   • The idea is to express a function as a series. The more terms given to the series, the closer the approximation.
   • Use Desmos to plot the graph of sinx. Then plot on the same graph the Maclaurin expansion to plot the expansion for the first term. For very small values of x this is accurate. Now plot until x^5. For relatively small values of x the graph should be identical to sinx. Finally plot it to x^15 (which is both good Maclaurin expansion plus shows what happens). Almost the entire graph shown will now be identical to sinx. The more terms found, the more accurate the approximation.
   • Not all functions can be written as a series, look at the presumptions to see why. It is also unwise to use this method for polynomials as you do not get an approximation that is at all accurate-merely a constant.
   1. Presumptions

      Nota:

      • f(x) can be expressed as a series 
      • The series can be differentiated term by term
      • f(x) and all derivatives exist at x=0
   2. Expressing f(x) as a series

      Nota:

      • f(x)=a+bx+cx^2+...+kx^r+...
      • a=f(0) b=f'(0) c=f''(0) etc.
      • Therefore: f(x)=f(0)+f'(0)x+f''(0)(x^2)/2!+...
2. Important Series

   Nota:

   • Most of these series are in the formula booklet.
   • 95% likelihood you WILL have to use this at some point in the exam. Make sure to be familiar both with the form and derivation. Either could be required. The form will not be worth many marks, but derivation could be, especially in a multi-part question. Be aware of how to derive them-it'll make it a lot quicker.
   1. e^x

      Nota:

      • f(x)=e^x f'(x)=e^x f''(x)=e^x
      • f(0)=1 f'(0)=1 f''(0)=1
      • e^x=1+x+x^2/2!+x^3/3!+...
   2. sinx

      Nota:

      • f(x)=sinx f'(x)=cosx f''(x)=-sinx f'''(x)=-cosx f''''(x)=sinx f'''''(x)=cosx
      • f(0)=0 f'(0)=1 f''(0)=0 f'''(0)=-1 f''''(0)=0 f'''''(0)=1
      • sinx=x-x^3/3!+x^5/5! etc.
   3. cosx

      Nota:

      • f(x)=cosx f'(x)=-sinx f''(x)=-cosx f'''(x)=sinx f''''(x)=cosx
      • f(0)=1 f'(0)=0 f''(0)=-1 f'''(0)=0 f''''(0)=1
      • cosx=1-x^2/2!+x^4/4!
   4. ln(1+x)

      Nota:

      • f(x)=ln(1+x) f'(x)=(1+x)^-1 f''(x)=-(1+x)^-2 f'''(x)=2(1+x)^-3
      • f(0)=0 f'(0)=1f''(0)=-1f'''(0)=2
      • ln(1+x)=x-x^2/2+x^3/3+...
   5. (1+x)^n

      Nota:

      • This should be familiar from the core units. It is just the binomial theorem. However, this may have to be proven in an FP3 exam, not just used
      • f(x)=(1+x)^n f'(x)=n(1+x)^(n-1) f''(x)=n(n-1)(1+x)^(n-2) f'''(x)=n(n-1)(n-2)(1+x)^(n-3)
      • f(0)=1 f'(0)=n f''(0)=n(n-1) f'''(0)=n(n-1)(n-2)
      • (1+x)^n=1+nx+n(n-1)x^2/2!+n(n-1)(n-2)x^3/3!+...
3. Important Limits

   Nota:

   • These are limits you need to remember and be able to derive
   1. x^ke^-x
   2. x^klnx
4. Limits

   Nota:

   • Limits are values which f(x) gets very very close to as x gets very close to a set value
   • They appear to a lesser extent in FP1, but in FP3 they come back in conjunction with Maclaurin Series
   • Limit questions tend to be in parts and you are often lead through them, finding the series expansion before finding the limit.
   • This is not true however for anything including the important limits mentioned here.
   1. Finding a simple limit
      1. Dividing by x^k

         Nota:

         • Dividing by x^k can turn two polynomials that look complex into a very simple limit.
         • Whether this is appropriate is a very case by case thing, but it typically is if you are finding a limit of a function in the form p(x)/q(x) where p(x) and q(x) are polynomials.
         • The example in the textbook summarises this well. Find the limit of f(x)=(1+x)/(1-2x) as x->infinity This limit cannot be found simply by using the limit in place of x. Instead dividing by x^k (in this case k=1 so x^k=x) is prudent. So f(x)=((1/x)+1)/((1/x)-2) Now we can use the limit. As x->infinity, 1/x tends to 0. Therefore the numerator tends to 1 and the denominator to -2. The limit is -1/2. 
      2. Obvious cases

         Nota:

         • These are very simple cases and will be worth only a handful of marks. However, they are still worth practicing.
         • These tend to be easy to see.
         • Example As x->0, (1+x)/(2-x)->1/2 because 1+x->1 and 2-x->2
         • Example As x->pi/2 sinx/(1-cosx)->1 because sinx->1 and cosx->0 as x->pi/2
   2. Using Series Expansion
5. Improper Integrals