Editing Taylor's theorem (section)

=== Estimates for the remainder ===

It is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it.  Suppose that ''f'' is {{nowrap|(''k'' + 1)}}-times continuously differentiable in an interval ''I'' containing ''a''. Suppose that there are real constants ''q'' and ''Q'' such that

<math display="block">q\le f^{(k+1)}(x)\le Q</math>

throughout ''I''. Then the remainder term satisfies the inequality<ref>{{harvnb|Apostol|1967|loc=§7.6}}</ref>

<math display="block">q\frac{(x-a)^{k+1}}{(k+1)!}\le R_k(x)\le Q\frac{(x-a)^{k+1}}{(k+1)!},</math>

if {{nowrap|''x'' > ''a''}}, and a similar estimate if {{nowrap|''x'' < ''a''}}. This is a simple consequence of the Lagrange form of the remainder. In particular, if

<math display="block">|f^{(k+1)}(x)|\le M</math>

on an interval {{nowrap|1=''I'' = (''a'' − ''r'',''a'' + ''r'')}} with some <math>r > 0</math> , then

<math display="block">|R_k(x)|\le M\frac{|x-a|^{k+1}}{(k+1)!}\le M\frac{r^{k+1}}{(k+1)!}</math>

for all {{nowrap|''x''∈(''a'' − ''r'',''a'' + ''r'').}} The second inequality is called a [[uniform convergence|uniform estimate]], because it holds uniformly for all ''x'' on the interval {{nowrap|(''a'' − ''r'',''a'' + ''r'').}}