Find the interval of convergence for each power series in Exercises 21–48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.
The interval of convergence for power series is .
The given power series is:
Let us assume therefore
The ratio for the absolute convergence is
Here the limit is So, by the ratio test of absolute convergence, we know that series will converge absolutely when that is .
Now, since the intervals are finite so we analyze the behavior of the series at the endpoints
The result is the alternating multiple of the harmonic series, which diverges.
The result is just a constant multiple of the harmonic series which converges conditionally.
Therefore, the interval of convergence of the power series is .
The second-order differential equation
where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by . It may be shown that is given by the following power series in x :
What is the interval of convergence for ?
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