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The function $ A $ defined by

$ A (x) = 1 + \frac {x^3}{2 \cdot 3} + \frac {x^6}{2 \cdot 3 \cdot 5 \cdot 6} + \frac {x^9}{\2 \cdot 3 \cdot 5 \cdot 6 \cdot 8 \cdot 9} + \cdot \cdot \cdot

is called an Airy function after the English mathematician and astronomer Sir George Airy (1801- 1892).

(a) Find the domain of the Airy function.

(b) Graph the first several partial sums on a common screen.

(c) If your CAS has built-in Airy functions, graph A on the same screen as the partial sums in part (b) and observe how the partial sums approximate A.

a) $(-\infty, \infty)$

b) see work for graph

c) see work for graph

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and this problem. We're focusing on power, Siri's and specifically how to find the interval of convergence or the domain of a power Siri's. So, for part A, I'm going to say Let A of X denote are some from an equals one to infinity of a seven. So, based on the problem that we're given in our textbook for n greater Than or equal toe one, a stipend plus one would be X to the third over three end minus one times three end times a seven. Now we want to get into a form. Remember that we can use some kind of technique with So we would get it in the form of a sub n plus one over ace of N, and we would have it to be X to the third over three and minus one times three n. So that means we can apply the ratio test. This is a format or a technique that we can use when we see something of this form. So we'll take the limit as an approaches infinity of the absolute value of ace, a ace of n plus one over the absolute value of a sub en so this would be the limit as an approaches infinity of the absolute value of X cubed over the absolute value of three in minus one times three in. So remember, we're having this this limit approach infinity. So this would be the same thing as the limit as an approaches infinity off the absolute value of of X cubed over infinity. So this limit would approach zero. And what that means is that because our limit of the absolute value of a seven plus one of the absolute value they said Ben was zero and that's less than one. We can say that the interval of convergence and the domain of this power, Siri's is the real numbers. So that would be negative. Pardon me? Negative infinity to positive infinity. So now the next part of this problem is we have to graph the 1st 12 partial sums. I just plugged it into a graphing utility, and this is what it came out toe look like. And then for part c, we're asked, what if we took the limit off these 12 partial sums and graft it? What would that look like? So this would be the graph of the limit of our partial son. So I hope that this problem helped you understand how we can analyze the power Siri's by possibly rearranging it so we can apply a power Siri's technique like the ratio test, and then put that into a graphing utility to understand the partial sums and visualize what are some or power Siri's would look like.

University of Denver