8.4. Taylor Series

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
f(x) =
where r(x) goes to zero as x goes to c.

Use this result and the function f(x)= to show that

The 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:

= 1 + x/2 + x r(x)

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:


which, together with the fact that r(x) = 0 will do the trick.

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