Question
The series \(1-x^{2}+\frac{x^{4}}{2!}-\frac{x^{6}}{3!}+\frac{x^{8}}{4!}+…..(-1)^{n}\frac{x^{2n}}{n!}+…\) converges to which of the following?
A \(\cos (x^{2})+\sin (x^{2})\)
B \(1 − xsinx\)
C \(cos x\)
D \(e^{-x^{2}}\)
Answer/Explanation
Question
Let f be a function such that \({f}'(x) = sin (x^2)\) and \(f (0) = 0\) What are the first three nonzero terms of the Maclaurin series for f ?
A \(x−\frac{x^5}{10}+\frac{x^9}{216}\)
B \(2x−x^5+\frac{x^9}{12}\)
C \(\frac{x^3}{3}−\frac{x^7}{21}+\frac{x^{11}}{55}\)
D \(\frac{x^3}{3}−\frac{x^7}{42}+\frac{x^{11}}{1320}\)
Answer/Explanation
Question
Find the Taylor series for \(f(x)=\frac{1}{\sqrt{x}}\) centered at x = 4.
(A)\(\sum_{n=0}^{\infty }(-1)^{n}\frac{1.3.5…(2n-1)}{2^{3n+1}n!}(x-4)^{n}\)
(B) \(\sum_{n=0}^{\infty }(-1)^{n}\frac{1.3.5…(2n+1)}{2^{3n+1}n!}(x-4)^{n}\)
(C) \(\sum_{n=0}^{\infty }(-1)^{n}\frac{1}{2^{2n+1}n}x^{n}\)
(D) \(\sum_{n=1}^{\infty }(-1)^{n}\frac{2.4.6….2n}{2^{3n+1}n!}(x-4)^{n}\)
Answer/Explanation
Ans:(A)
Question
Find the Taylor series for \(f(x)=\cos (\pi x)\) centered at x = 3
(A) \(\sum_{n=0}^{\infty }\frac{(-1)^{n}\pi ^{2n-2}}{(2n)!}(x-1)^{2n}\)
(B)\(\sum_{n=0}^{\infty }\frac{(-1)^{n+1}\pi ^{2n}}{(2n)!}(x-1)^{2n}\)
(C) \(\sum_{n=0}^{\infty }\frac{\pi ^{2n-1}}{(2n-1)!}x^{2n-1}\)
(D) \(\sum_{n=0}^{\infty }\frac{(-1)^{n+1}}{n!\pi ^{2n}}(x-1)^{n}\).
Answer/Explanation
Ans:
