Question

Function has the property that f^(n) (x) = f^(n + 2) (x) : n ≥ 1 : n ∈ N Then which of the following is the expression for f(x) in most general form.

(a) x+\(\frac{x^3}{3!}+\frac{x^5}{5!}+…..\infty\)

(b) 1+\(\frac{x^2}{2!}+\frac{x^4}{4!}+….\infty\)

(c) c1 \( \times cosh(x)+c_2\times sinh(x)\)

(d) cosh(x) + sinh(x)

The question was posed to me in exam.

This question is from Taylor Mclaurin Series topic in portion Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (c) c1 \( \times cosh(x)+c_2\times sinh(x)\)

Best explanation: Consider the general form of Mclaurin series for f(x)

f(x)=a0+a1x+a2x^2+a3x^3+….∞

Then we have

f^(0)(0)=f^(n+2)(0)

n!an=(n+2)!an+2

an+2=\(\frac{a^n}{(n+1)(n+2)}\)

a2n=\(\frac{a_0}{(2n)!}\)

OR

a2n+1=\(\frac{a_1}{(2n+1)!}\)

One can deduce the general expression for f(x) to be

f(x)=\(a_0 \times (\frac{1}{1} + \frac{x^2}{2!} + …\infty) + a_1 \times (\frac{x}{1!}+\frac{x^3}{3!}+\frac{x^5}{5!}+…\infty)\)

Now writing a0 = c1 and a1 = c2 we have

f(x) = c1 * cosh(x) + c2 * sinh(x).