Here is quiz 8.
Problem 1 asks to find the power series for .
Method 1: Using basic algebra, expand the cube:
This gives as a sum of powers of . Since there is only one way of doing this, this is the series we were looking for (and we have ).
Method 2: Let . Then we have:
Problem 2 asks to find the power series for .
Perhaps the easiest way to proceed is to note that
Problem 3 asks to find the radius of convergence of the series
As usual, we look at and , which we find by replacing with in the previous formula, so we have
We then take the quotient and simplify:
and this last expression converges to as . To see this, note that
By the ratio test, the series converges if and diverges if . Note that is the same as or i.e., .
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There is a small typo in the pdf file: In method 1 for problem 1, “Since here” should be “Since there”.