How do you find the maclaurin series expansion of #x^3/(1+x^2)#?

Answer 1

Use the Maclaurin series for #1/(1-t)# and substitution to find:

#x^3/(1+x^2) = sum_(n=0)^oo (-1)^n x^(2n+3)#

The Maclaurin series for #1/(1-t)# is #sum_(n=0)^oo t^n#
since #(1-t) sum_(n=0)^oo t^n = sum_(n=0)^oo t^n - t sum_(n=0)^oo t^n = sum_(n=0)^oo t^n - sum_(n=1)^oo t^n = t^0 = 1#
Substitute #t = -x^2# to find:
#1/(1+x^2) = sum_(n=0)^oo (-x^2)^n = sum_(n=0)^oo (-1)^n x^(2n)#
Then multiply by #x^3# to get:
#x^3/(1+x^2) = x^3 sum_(n=0)^oo (-1)^n x^(2n) = sum_(n=0)^oo (-1)^n x^(2n+3)#
This is a geometric series with common ratio #-x^2#, hence the radius of convergence is #1#.
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Answer 2

To find the Maclaurin series expansion of ( \frac{x^3}{1+x^2} ), follow these steps:

  1. Start with the function ( \frac{x^3}{1+x^2} ).
  2. Express ( \frac{x^3}{1+x^2} ) as a geometric series by using the formula for the sum of an infinite geometric series.
  3. Find the Maclaurin series expansion of the terms in the geometric series.
  4. Substitute the Maclaurin series expansions into the expression for the geometric series.
  5. Simplify the resulting expression to obtain the Maclaurin series expansion of ( \frac{x^3}{1+x^2} ).

The Maclaurin series expansion of ( \frac{x^3}{1+x^2} ) will involve powers of ( x ) and their coefficients, determined by the derivatives of the function evaluated at ( x = 0 ).

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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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