How do you find the maclaurin series expansion of #f(x) = x/(1-x)#?

Answer 1

If #|x|<1#, the expansion is #x+x^2+x^3+...=sum_{i=1}^infty x^i#

This is already a known expansion, since #sum_{i=1}^infty x^i=x/(1-x)#.

You must compute the derivatives, evaluate them in zero, and then apply the expression to understand why this is the case.

#f(x)= f(0) + f'(0)* x + (f''(0)) /(2!) * x^2 + (f'''(0)) /(3!)x^3+... #
and verify that every coefficient #f^{(n)}(0)/(n!)# equals one.

That's all there is to it if your main concern is how you had to discover the expansion; if you would like verification of the last point, please ask.

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Answer 2

To find the Maclaurin series expansion of ( f(x) = \frac{x}{1-x} ), we'll start by expressing it as a geometric series.

[ f(x) = x \cdot \frac{1}{1-x} ]

Now, we'll use the geometric series formula:

[ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n ]

Multiplying by ( x ):

[ x \cdot \frac{1}{1 - x} = x \cdot \sum_{n=0}^{\infty} x^n ]

[ = \sum_{n=0}^{\infty} x^{n+1} ]

This series represents ( f(x) ) in terms of its Maclaurin series expansion. However, we need to make sure that it converges. The radius of convergence ( R ) for this series is 1, which means it converges for ( |x| < 1 ).

So, the Maclaurin series expansion of ( f(x) = \frac{x}{1-x} ) is:

[ \sum_{n=0}^{\infty} x^{n+1} ]

This can also be written as:

[ \sum_{n=1}^{\infty} x^n ]

This series converges for ( |x| < 1 ).

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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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