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

Answer 1

#x^p = sum_(k=0)^oo ((p),(k)) (x-1)^k#

where # ((p),(k)) # is the generalized binomial coefficient.

We have to distinguish two cases:

(1) If # p in NN# then the derivatives of order grater than #p# of #f(x)# are identically null. In such case:
#(df)/dx = px^(p-1)#

...

#(df^k)/dx^k = p(p-1)...(p-k+1)x^(p-k) = (p!)/(p-k!)x^(p-k)#
and for #x=1#
#[(df^k)/dx^k ]_(x=1) = (p!)/(p-k!)#

so the Taylor series has a finite number of terms:

#x^p = sum_(k=0)^p (p!)/(p-k!)(x-1)^k/(k!) = sum_(k=0)^p ((p),(k)) (x-1)^k#
where #((p),(k)) = (p!)/((k!)(p-k)!)# is a binomial coefficient.
(2) If # p notin NN# then we can use the definition of generalized binomial coefficient:
#((p),(k)) = (p(p-1)(p-2)...(p-k+1))/(k!)#
where #p in RR# and #k in NN#

In this case the derivatives of all orders are:

#(df^k)/dx^k = p(p-1)...(p-k+1)x^(p-k) = ((p),(k)) k! x^(p-k)#
and for #x=1#:
#[(df^k)/dx^k ]_(x=1) = ((p),(k)) (k!)#

so the Taylor series is:

#x^p = sum_(k=0)^oo ((p),(k)) (k!)(x-1)^k/(k!) = sum_(k=0)^oo ((p),(k)) (x-1)^k#

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

#((p),(k)) = (p(p-1)(p-2)...color(blue)((p-p))..(p-k))/(k!) = 0#
for #p,k in NN# and #k > p#

then we can unify the notation as:

#x^p = sum_(k=0)^oo ((p),(k)) (x-1)^k#
for any #p#.
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Answer 2

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 ]

  1. Compute ( f(a) = f(1) = 1^p = 1 ).
  2. Compute ( f'(x) = px^{p-1} ) and evaluate at ( x = 1 ): ( f'(1) = p ).
  3. Compute ( f''(x) = p(p-1)x^{p-2} ) and evaluate at ( x = 1 ): ( f''(1) = p(p-1) ).
  4. 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 ]

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Answer from HIX Tutor

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.

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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