Question

The expansion of e^Sin(x) is?

(a) 1 + x + ^x^2⁄2 + ^x^4⁄8 +….

(b) 1 + x + ^x^2⁄2 – ^x^4⁄8 +….

(c) 1 + x – ^x^2⁄2 + ^x^4⁄8 +….

(d) 1 + x + ^x^3⁄6 – ^x^5⁄10 +….

I have been asked this question in examination.

My query is from Taylor Mclaurin Series topic in division Differential Calculus of Engineering Mathematics

Answer 1

Right answer is (b) 1 + x + ^x^2⁄2 – ^x^4⁄8 +….

Best explanation: Now f(x) = e^Sin(x), f(0) = 1

Hence, f^‘ (x)=f(x)Cos(x), f^‘ (0) = 1

f^” (x)=f^‘ (x)Cos(x) – f(x)Sin(x),f^” (0)=1

f^”’ (x)=f^” (x)Cos(x) – 2f^‘ (x)Sin(x) – f(x) cos⁡(x),f^”’ (x) = 0

f^”” (x)=f^”’ (x)Cos(x) – 3f^” (x)Sin(x) – 3f^‘ (x) cos⁡(x) + f(x) sin⁡(x), f^”” (x) = -3

Hence,

f(x) = e^Sin(x) = 1 + x + ^x^2⁄2 – ^x^4⁄8 +…. (By mclaurin’ sexpansion)