Advanced High-School Mathematics

298 CHAPTER 5 Series and Differential Equations

Step 4. Use step 1 to write the above as

m(2m−1)

3

<

(2m+ 1)^2

π^2

∑m

k=1

1

k^2

< m+

m(2m−1)

3

Step 5. Multiply the inequality of step 4 through by

π^2

4 m^2

and let

m→∞.What do you get?

5.4.2 Error analysis and Taylor’s theorem

In this final subsection we wish to address two important questions:

Question A: If Pn(x) is the Maclaurin (or Taylor) polynomial of

degree n for the function f(x), how good is the approximation

f(x)≈Pn(x)? More precisely, how large can the error

|f(x)−Pn(x)|be?

Question B:When can we say that the Maclaurin or Taylor series of

f(x) actually converges tof(x)?

The answers to these questions are highly related.

The answer to both of these questions is actually contained inTay-
lor’s Theorem with Remainder. Before stating this theorem, I
want to indicate that this theorem is really just a generalization of the
Mean Value Theorem, which I’ll state below as a reminder.

Mean Value Theorem. Let f be a differentiable function on some
open intervalI. Ifaandxare both inI, then there exists a real number
cbetweenaandxsuch that

f(x)−f(a)

x−a

=f′(c).

Put differently, there exists a numbercbetweenaandxsuch that

f(x) =f(a) +f′(c)(x−a).