Educator

Taylor Polynomial Applications

Main formulas:

Alternating Series Error Bound: If ∑a_n is a series that satisfies Alternating Series Test, and we use the partial sum s_n as an estimate of the total sum S (the true answer), then our error E could be positive or negative, but it’s bounded by |a_n+1|:

|E| ≤ |a_n+1|

Taylor’s Remainder Theorem: Suppose you use the Taylor polynomial T_k (x) around a to estimate f (x) at a value of x near a. Find the maximum value of f (k+1) on the interval between a and x (or an upper bound for it). Call that M. Then the

Error ≤ .

Hints and tips:

• The point of these formulas is that (among other applications) we will use Taylor polynomials to estimate the values of f (x) for some functions that would otherwise be difficult to compute. Then we will use these error formulas to determine how accurate our estimates are.

• General principles on accuracy of Taylor Series:

1. If they don’t converge, then they’re no good at all.

2. But they usually converge, since 1/n! gets small quickly.

3. They are very accurate when x is close to the center a, since (x − a)ⁿ gets small quickly. When x is far from a, (x − a)ⁿ gets big and they aren’t very accurate.

4. This is why we use different center values of a depending on what values of x we want to use the series for.

• When you use Taylor Series to estimate values of f (x), you need to center the series around a value of a for which f (a), f ′ (a), f ′′ (a), and so on, will be easy to compute.

• In Taylor’s Remainder Theorem, the difficult part is often coming up with the constant M . Here are several points to remember:

• Since M only needs to be an upper bound, not an exact value, it is OK to estimate any quantities in your expression for | f ^(k+1) |. However, it is important to be certain that your estimate is higher than the true value, not lower.

• |sin x| ≤ 1, and similarly for cos x.

• |a − b| ≤ |a| + |b|.

• To estimate high, make the numerator as big as possible and the denominator as small as possible.

• Sometimes we are given the error tolerance ahead of time and we must use these error formulas to reverse-engineer the number of terms we need to be accurate to within the given tolerance. (For example, an engineer building a bridge might know that she needs calculations accurate to one ten-thousandth of a meter. Or a calculator might have a display of ten decimal places, so the manufacturer must design the internal algorithms to produce answers with ten digits of accuracy.)

• In both formulas (but especially for Taylor’s Remainder Theorem), if you are given the error tolerance, it may be hard to solve the formula explicitly for n or k. You may need to use trial and error to find the right n or k. (Remember that they are whole numbers.)

• A common mistake is to think that the Taylor polynomial T_k (x) has k terms. k refers to the degree, not the number of terms. So, for example, the Taylor polynomial T₄ (x) for f (x) = sin x centered around a = 0 is T₄ (x) = x − x³&frasl 6 , because the term of x⁴ is zero.