Question

Lagrange’s Remainder for Maclaurin’s Theorem is given by _____________

(a) \(\frac{x^n}{(n-1)!}f^{(n)}(θx) \)

(b) \(\frac{x^n}{n!} f^{(n)}(θx)\)

(c) \(\frac{x^{n-1}}{n!} f^{(n)}(θx)\)

(d) \(\frac{x^n}{n!}f^{(n-1)}(θx)\)

This question was posed to me in a national level competition.

The doubt is from Generalized Mean Value Theorem topic in chapter Differential Calculus of Engineering Mathematics

Answer 1

Correct option is (b) \(\frac{x^n}{n!} f^{(n)}(θx)\)

The best I can explain: Maclaurin’s Theorem is a special case of Taylor’s Theorem; hence Schlomilch’s Remainder for Maclaurin’s Theorem is given by, \(\frac{x^n(1-θ)^{n-p}}{(n-1)!p} f^{(n)}(θx).\) To obtain Lagrange’s Remainder for Maclaurin’s Theorem, we put p=n, which gives us, \(\frac{x^n}{n!} f^{(n)}(θx).\)