Question

Let τ(f(x)) denote the Taylor series for some function f(x). Then the value of τ(τ(τ(f(1729)))) – 2τ(τ(f(1729))) + τ(f(1729)) is?

(a) 1729

(b) -1

(c) 1

(d) 0

The question was asked in homework.

My question comes from Taylor Mclaurin Series in chapter Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (d) 0

Best explanation: We know that the Mclaurin Series for any given function always yields a polynomial (finite OR infinite).

Further the Mclaurin series of this polynomial (i.e.τ(τ(f(x)))) is also a polynomial. Due to uniqueness of this polynomial, no matter how many nested Mclaurin series we might find, they are all equal. Thus, we have

τ(τ…….(f(x))….)) = f(x)

Substituting this into our required expression we have

 = f(1729) – 2f(1729) + f(1729)

 = 0.