The Taylor Series


The Remainder Term

We now use integration by parts to determine just how good of an approximation is given by the Taylor polynomial of degree n , p n(x) . By the fundamental theorem of calculus,

f (b) - f (a) = f'(t)dt    

Integrating by parts, choosing - (b - t) as the antiderivative of 1 , we have


f'(t)dt = - f'(t)(b - t)|a b + f (2)(t)(b - t)dt  
  = f'(a)(b - a) + f (2)(t)(b - t)dt  

Again, integrating by parts yields


f (2)(t)(b - t)dt = - f (2)(t) + f (3)(t) dt  
  = f (2)(a) + f (3)(t) dt  

Putting these equations together, we have

f (b) = f (a) + f'(a)(b - a) + f (2)(a) + f (3)(t) dt    

Continuing the process, we arrive at

f (b) = f (a) + f'(a)(b - a) + ... + f (n-1)(a) + f (n)(t) dt    

Substituting x for b , we have an expression for f (x) , called Taylor's formula at x = a , involving the familiar Taylor polynomial of degree n - 1 for f and an integral called the remainder term and denoted by r n(x) :


f (x) = f (a) + f'(a)(x - a) + ... + f (n-1)(a) + f (n)(t) dt  
  = p n-1(x) + r n(x)  

Therefore, in order to compute how close p n(x) is to f (x) , we need to find the magnitude of the remainder term. Fortunately, there is a simpler way to express r n(x) .

Letting x be fixed for a moment, choose numbers m and M in the interval [a, x] so that f (m) is the minimum value of f on the interval and f (M) the maximum value. Then for any t in [a, x] ,

f (n)(m)f (n)(t)f (n)(M)    

The corresponding integrals must satisfy similar inequalities:

f (n)(m) dtr n(x)≤f (n)(M) dt    

or

f (n)(m)r n(x)≤f (n)(M)    

By the intermediate value theorem,

f (n)(t)    

takes on all values between its minimum and maximum in the interval [a, x] , so there exists some c in [a, x] such that

r n(x) = (x - a)n    

The expression on the right looks very much like the n th term of the Taylor polynomial--the only difference is that the derivative is evaluated at some number c in the interval [a, x] rather than at a .

Now that the remainder term is in a more manageable form, we can try to bound it. Suppose we have a bound, B n , for the absolute value of the n th derivative of f on the interval [a, x] . That is,

| f (n)(c)|≤B n    

for all c in [a, b] . Then we have the bound

| r n(x)|≤| x - a|n    

To conclude, we restate the results of this section in the case a = 0 . The Taylor formula in this case reads

f (x) = f (0) + f'(0)x + ... + f (n-1)(0) + f (n)(t) dt    

with remainder term bounded by

| r n(x)|≤| x|n    

Take a Study Break

SparkLife

Star Trek gets SEXY

Chris Pine and Zoe Saldana heat up the red carpet!

SparkLife

Are you afraid of relationships?

Auntie SparkNotes can help!

SparkLife

Wanna get JLaw's gorgeous glow?

Click here for simple, sexy makeup tricks!

SparkLife

Sexy starlet style

See every single look from the Met Gala!

SparkLife

Who'd be on your zombie-apocalypse crew?

We already dib'sed Genghis Khan.

Geek out!

The MindHut

Geeky Actresses: Then and Now

Before the fame!

The MindHut

9 Scientific Inaccuracies in Iron Man 3

Click to see what they got wrong.

The MindHut

Top 10 Predictions Sci-Fi Got WRONG

So wrong, they're WRONG.

The MindHut

The 15 Most Awesome Robots, Ever

These Robots Rock!

The MindHut

If You Like Game of Thrones...

...Then you'll LOVE these books!

The Book

Cover image

Read What You Love, Anywhere You Like

Get Our FREE NOOK Reading Apps