How do you find the product of power series?

Answer 1
#(sum_{n=0}^infty a_nx^n)cdot(sum_{n=0}^infty b_nx^n)=sum_{n=0}^infty(sum_{k=0}^na_kb_{n-k})x^n#

Let us look at some details.

#(sum_{n=0}^infty a_nx^n)cdot(sum_{n=0}^infty b_nx^n)#

by writing out the first few terms,

#=(a_0+a_1x+a_2x^2+cdots)cdot(b_0+b_1x+b_2x^2+cdots)#

by collecting the like terms,

#=a_0b_0x^0+(a_0b_1+a_1b_0)x^1+(a_0b_2+a_1b_1+a_2b_0)x^2+cdots#

by using sigma notation,

#=sum_{n=0}^infty(sum_{k=0}^na_kb_{n-k})x^n#
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Answer 2

To find the product of two power series, follow these steps:

  1. Write the two power series as infinite sums.
  2. Multiply each term of the first series by each term of the second series.
  3. Combine like terms and simplify the resulting expression.
  4. If necessary, identify the coefficients of the resulting power series.

Note: Depending on the specific form of the power series, additional techniques may be required for simplification, such as manipulating the series algebraically or using known series identities.

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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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