# What is the Remainder Term in a Taylor Series?

Taylor's residual term

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The remainder term in a Taylor series represents the error or the difference between the actual value of a function and its approximation using a finite number of terms from the Taylor series expansion. It is often denoted by ( R_n(x) ) and is expressed as the difference between the function and the nth-degree Taylor polynomial divided by ( (n+1) ) factorial, multiplied by the ( (n+1) )st derivative of the function at a certain point ( c ), raised to the power of ( (n+1) ), and multiplied by ( (x-c)^{n+1} ), where ( c ) lies between the point of expansion and the point at which the approximation is being made. Mathematically, the remainder term can be written as:

[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-c)^{n+1} ]

The remainder term allows us to estimate how close the approximation using a finite number of terms in the Taylor series is to the actual function value.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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