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How do you find the power series representation for the function #f(x)=1/(1-x)# ?

Answer 1

Evelyn Agard

Remember that the sum of the geometric series can be found by #a/{1-r}=sum_{n=0}^infty ar^n#.

By setting #a=1# and #r=x#, #1/{1-x}=sum_{n=0}^infty x^n#.

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Answer 2

Isaiah Allen

To find the power series representation for the function ( f(x) = \frac{1}{1-x} ), you can use the geometric series formula. The geometric series formula states that for ( |x| < 1 ),

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

This is a well-known result derived from the geometric series formula. So, the power series representation for ( f(x) = \frac{1}{1-x} ) is

[ f(x) = \sum_{n=0}^{\infty} x^n ]

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.