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# Determine the interval of convergence for a given power series and show that it satisfies a given differential equation

Consider the function defined by the power series

Determine the interval of convergence for and show that it satisfies the differential equation

First, to determine the interval of convergence for the power series we use the ratio test

Hence, the series converges for all . Next, to show that it satisfies the given differential equation we take the first two derivatives,

Then, we have

Thus, indeed satisfies the given differential equation.

Further, in a previous exercise (Section 8.14, Exercise #2) that the solution of the differential equation are all of the form

For this problem we also have so

Finally, we know this function is an even function (since for all because we have inside the sum is the only term). This means we must have . Hence, we must have