Question

How do you find the remainder term `R_3(x;1)` for `f(x)=sin(2x)`?

Answer 1

Remainder Term of Taylor Series

`R_n(x;c)={f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}`,

where `z` is a number between `x` and `c`.

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Let us find `R_3(x;1)` for `f(x)=sin2x`.

By taking derivatives,

`f'(x)=2cos2x`
`f''(x)=-4sin2x`
`f'''(x)=-8cos2x`
`f^{(4)}(x)=16sin2x`

So, we have

`R_3(x;1)={16sin2z}/{4!}(x-1)^4`,

where `z` is a number between `x` and `1`.

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I hope that this was helpful.