Answer to Question #271768 in Calculus for Sara

Question #271768

How to calculate cos x from its nth term in Maclaurin series??


1
Expert's answer
2021-11-29T05:26:11-0500

The Maclaurin series of "f(x)=\\cos x" is

 

"f(x)=\\sum_{n=0}^{\\infty}(-1)^{n} \\frac{x^{2 n}}{(2 n) !}"

 

Let us look at some details.

The Maclaurin series for f(x) in general can be found by

"f(x)=\\sum_{n=0}^{\\infty} \\frac{f^{(n)}(0)}{n !} x^{n}"

Let us find the Maclaurin series for "f(x)=\\cos x."

By taking the derivatives,

 "\\begin{aligned}\n\n&f(x)=\\cos x \\Rightarrow f(0)=\\cos (0)=1 \\\\\n\n&f^{\\prime}(x)=-\\sin x \\Rightarrow f^{\\prime}(0)=-\\sin (0)=0 \\\\\n\n&f^{\\prime \\prime}(x)=-\\cos x \\Rightarrow f^{\\prime \\prime}(0)=-\\cos (0)=-1 \\\\\n\n&f^{\\prime \\prime \\prime}(x)=\\sin x \\Rightarrow f^{\\prime \\prime \\prime}(0)=\\sin (0)=0 \\\\\n\n&f^{(4)}(x)=\\cos x \\Rightarrow f^{(4)}(0)=\\cos (0)=1\n\n\\end{aligned}"

Since "f(x)=f^{(4)}(x)," the cycle of "\\{1,0,-1,0\\}" repeats itself.

So, we have the series

"f(x)=1-\\frac{x^{2}}{2 !}+\\frac{x^{4}}{4 !}-\\cdots=\\sum_{n=0}^{\\infty}(-1)^{n} \\frac{x^{2 n}}{(2 n) !}"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS