Infinite Sequences and Series: Multiplication and Division of Power Series

In this video I go over a very in-depth tutorial on division and multiplication of power series just as we would for polynomials. In my earlier video I had gone over an example on multiplying and dividing power series using the basic polynomial expansion and polynomial long division techniques, but I had not discussed the reasoning for why it works. In this video, however, I derive the power series that results from multiplying two power series and show that the resulting coefficients involve a “convolution” summation that can also be obtained in the same way had we just multiplied the power series as if they were polynomials.

The division of power series is effectively just the reverse of multiplication and thus to divide a power series is the same as figuring out which power series multiplies to get the numerator. I show that polynomial long division and double checking using multiplication yield the same result as expected.

Also in this video, I go over briefly on Euler’s identity and imaginary numbers to illustrate, in Example 3, that if we consider imaginary numbers then we need to account for when the denominator of a power series is equal to zero in the imaginary or complex set of numbers. This particular example was in regard to obtaining the power series of hyperbolic cosecant or csch.

The topics covered in this video are listed below with their time stamps.

• @ 1:14 - Topics to Cover

1. @ 2:19 - Recap on Power Series
   • @ 5:54 - Taylor and Maclaurin Series
2. @ 6:59 - Recap on Earlier Example on Multiplication and Division of Power Series
3. @ 18:21 - Multiplication of Power Series
   • @ 49:43 - Example 1: e^x/(1-x)
   • @ 1:03:36 - Example 2: sin^2(x)
4. @ 1:27:37 - Division of Power Series
   • @ 1:34:04 - Example 3: csch(x)
     • @ 1:53:20 - Imaginary Numbers and Euler's Identity
     • @ 2:02:08 - Double Checking Division By Multiplication

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