Interactive Real Analysis
- part of Example 8.4.9: Applying the Lagrange Remainder
Show that if f is n-times continuously differentiable
on [a, b] and c 
[a, b],
then
Back
[a, b],
then
f(x) =where r(x) goes to zero as x goes to c.
Use this result and the function
f(x)=
to show that
BackThe first statement is a straight-forward application of the Lagrange remainder theorem - try it youself!
As for the application, let
f(x)=, which is continuously
differentiable around c = 0. According to our statement (and taking
the first derivative at zero) we have:
for some r(x) with
r(x) = 0. To
apply this to our problem, we need to see
involved somehow. Therefore we factor an n to get:
Now we can apply our approximation to the second root, with
x = 1/
, to get:
Therefore
which, together with the fact that
r(x) = 0
will do the trick.