How do you use the binomial theorem to find the Maclaurin series for the function #y=f(x)# ?

Answer 1

Binomial Series

#(1+x)^{alpha}=sum_{n=0}^infty((alpha),(n))x^n#,
where #((alpha),(n))={alpha(alpha-1)(alpha-2)cdot cdots cdot(alpha-n+1)}/{n!}#.

Let us look at this example below.

#1/{sqrt{1+x}}#

by rewriting a bit,

#=(1+x)^{-1/2}#

by Binomial Series,

#=sum_{n=0}^infty((-1/2),(n))x^n#

by writing out the binomial coefficients,

#=sum_{n=0}^infty{(-1/2)(-3/2)(-5/2)cdots(-{2n-1}/2)}/{n!}x^n#

by simplifying the coefficients a bit,

#=sum_{n=0}^infty(-1)^n{1cdot3cdot5cdot cdots cdot(2n-1)}/{2^n n!}x^n#

I hope that this was helpful.

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Answer 2

To use the binomial theorem to find the Maclaurin series for a function ( y = f(x) ), follow these steps:

  1. Express ( f(x) ) in a form that resembles ((1 + x)^n) where ( n ) is a real number.
  2. Use the binomial theorem, which states that ((1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k), to expand ( f(x) ) into a series.
  3. Adjust the series to center it at ( x = 0 ) if it's not already centered there. This will give you the Maclaurin series.

The Maclaurin series for ( f(x) ) is essentially the Taylor series centered at ( x = 0 ).

Remember, not all functions can be represented by a simple form like ((1 + x)^n) for direct application of the binomial theorem. In such cases, you may need to manipulate the function or use other series expansions to find its Maclaurin series.

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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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