# Set up expansion formula for Taylor Series #f(x)=\root(3)(x)# centered round #a=1#?

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So far I have the exponent as #x^(-2/3-n)# , and the denominator as #3# . But the rest in the numerator?

What kind of factorial is #(1*2*5*8*....)# ?

So far I have the exponent as

What kind of factorial is

##### 1 Answer

#### Explanation:

We see that the derivatives alternate signs after the first term and the

Furthermore, the exponent can be represented by

The denominator involves

For the numerator, a distinct pattern emerges if we disregard

It can't be represented by a factorial (or even double factorial), but we can use the following notation to indicate the terms all have differences of

We have

So,

Thus, using the standard form of a Taylor Series about

However, if we take the above summation at