Power Series: A power series is just an infinite series that has a variable (x) in the summation terms, with x being raised to integer "powers" (hence the name). Writing a function in this format opens up all kinds of possibilities when it comes to integrating and differentiating (and for you engineers, solving differential equations).
The format of power series is really just that of a glorified polynomial, except the polynomial doesn't end; it just keeps growing longer and longer...an infinite number of terms.
So, since a power series involves adding up an infinite number of terms, and since we know that the sum of an infinite number of terms can converge or diverge, a compelling question is "For what x-values will a given power series converge?"
We determine that by finding what's called "The Interval of Convergence". Just as with the domain of a function, the Interval of Convergence" will tell us which values of x are valid to use in our new Power Series Function.
Videos:
Example: Finding the Interval of Convergence
Tip: Stop now and do the Interval of Convergence problems from the text (see below).
New Series from Old: Now that you've seen how a Geometric Series can be reconfigured to actually be a function, involving x, we're going to investigate how we can use substitution to develop new series based on one we actually know (or one that is given to us).
Video:
New Series from Old (Note: The intro won't make total sense until Section 10.3, but it will show you how to do the problems.)
Tip: Stop now and do the "New series from old" problems from the text (see below).
Assigned Problems from Textbook
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1st Edition Section 10.2, page 703 Interval of Convergence: 9, 11, 17 New series from old: 29, 32, 31, 35, 37
3rd Edition Section 11.2, page 729 Interval of Convergence: 9, 10, 15 New series from old: 41, 42, 43, 47, 49 |