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Question

Give the Maclaurin series for sinx.


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Solution

Step 1: Maclaurin series explanation

A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function.

The Maclaurin series of a function f(x) up to order n may be found using series [f,x,0,n].

It is a special case of the Taylor series when x=0.

The general equation of the Maclaurin series is

f(x)=f(x0)+f'(x0)(x-x0)+f(x0)2!(x-x0)2+f'(x0)3!(x-x0)3+..

Step 2: Maclaurin series for sinx

Given, fx=sinx

Using x=0, the given equation function becomes

f(0)=sin(0)=0

Now taking the derivatives of the given function and using x=0, we have

  1. f(0)=cos(0)=1
  2. f''(0)=sin(0)=0=0
  3. f'''(0)=sin(0)=0
  4. f''''(0)=sin(0)=0

Therefore, we get the series as,

f(x)=f(0)+xf'(0)+x22!f''(0)+x33!f'''(0)+x44!f''''(0)+.

Putting the values in the above series, we get

sinx=1-x22!+x44!-x66!+

Hence, the Maclaurin series for sinx is 1-x22!+x44!-x66!+.


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