Taylor Series and Partial Fraction Decomposition

This video tutorial covers power series for rational functions, starting with an introduction to power series and polynomial expansion. It explains how to express polynomials as power series around a point and demonstrates the use of Taylor series to find coefficients. The tutorial explores series for rational functions, including convergence and geometric series, and shows how to expand rational functions using power series. It also covers determining coefficients in power series expansions and using partial fraction decomposition for series.

What is a polynomial in terms of power series?

• A polynomial is an infinite power series.

• A polynomial is a finite power series.

• A polynomial is a power series with non-zero coefficients.

• A polynomial is a power series with infinite coefficients.

How can you expand a polynomial as a power series around a point?

• By writing it as a polynomial in the quantity x divided by x sub zero.

• By writing it as a polynomial in the quantity x times x sub zero.

• By writing it as a polynomial in the quantity x minus x sub zero.

• By writing it as a polynomial in the quantity x plus x sub zero.

What is the first step in using the Taylor series formula for a polynomial?

• Determine the second derivative at the point.

• Determine the first derivative at the point.

• Determine the function value at the point.

• Determine the third derivative at the point.

What does the geometric power series converge to?

• The function 1 divided by the quantity 1 plus X.

• The function X divided by the quantity 1 minus X.

• The function X divided by the quantity 1 plus X.

• The function 1 divided by the quantity 1 minus X.

What is the radius of convergence for the geometric power series?

• The radius of convergence is 0.

• The radius of convergence is 1.

• The radius of convergence is 2.

• The radius of convergence is infinite.

How do you write a power series for a rational function using geometric series?

• By replacing x with one in the power series for 1 divided by the quantity 1 minus X.

• By replacing x with positive x in the power series for 1 divided by the quantity 1 minus X.

• By replacing x with negative x in the power series for 1 divided by the quantity 1 minus X.

• By replacing x with zero in the power series for 1 divided by the quantity 1 minus X.

What is the pattern for the coefficients C sub k in the power series expansion?

• C sub k is always zero.

• C sub k is always positive.

• C sub k alternates between positive and negative.

• C sub k is always negative.

What method can be used for more complex rational functions to find the Taylor series?

• Integration by parts.

• Polynomial division.

• Geometric series expansion.

• Partial fraction decomposition.

In partial fraction decomposition, how is the rational function expressed?

• As a quotient of simpler rational functions.

• As a sum of simpler rational functions.

• As a product of simpler rational functions.

• As a difference of simpler rational functions.

What is the final result of the partial fraction decomposition example given?

• Positive X plus the sum from k equals three to infinity.

• Zero plus the sum from k equals three to infinity.

• Negative X plus the sum from k equals three to infinity.

• One plus the sum from k equals three to infinity.