# How do you find the Taylor's formula for #f(x)=x^p# for x-1?

where

We have to distinguish two cases:

...

so the Taylor series has a finite number of terms:

In this case the derivatives of all orders are:

so the Taylor series is:

Note that if we use the definition of generalized binomial coefficient, we have:

then we can unify the notation as:

By signing up, you agree to our Terms of Service and Privacy Policy

To find Taylor's formula for the function ( f(x) = x^p ) centered at ( x = 1 ), we use Taylor's formula:

[ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots ]

- Compute ( f(a) = f(1) = 1^p = 1 ).
- Compute ( f'(x) = px^{p-1} ) and evaluate at ( x = 1 ): ( f'(1) = p ).
- Compute ( f''(x) = p(p-1)x^{p-2} ) and evaluate at ( x = 1 ): ( f''(1) = p(p-1) ).
- Compute ( f'''(x) = p(p-1)(p-2)x^{p-3} ) and evaluate at ( x = 1 ): ( f'''(1) = p(p-1)(p-2) ), and so on.

The Taylor series for ( f(x) = x^p ) centered at ( x = 1 ) is:

[ f(x) = 1 + p(x - 1) + \frac{p(p-1)}{2!}(x - 1)^2 + \frac{p(p-1)(p-2)}{3!}(x - 1)^3 + \cdots ]

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you find the Maclaurin Series for #Sin(x^2)#?
- How do you find the maclaurin series expansion of #f(x)=cos(5x^2)#?
- How do you find the power series for #f(x)=1/(1+x^2)^2# and determine its radius of convergence?
- How do you use a Power Series to estimate the integral #int_0^0.01e^(x^2)dx# ?
- Assuming that the range of #sin^(-1)x # is #(-oo, oo)#, is # x sin^(-1) x # differentiable, for #sin^(-1)x in [0, 2pi]#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7