Chapter 8: Q. 55 (page 692)
Find the Maclaurin series for the functions in Exercises 51–60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
Soltution will be provided later
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Find the interval of convergence for power series:
Let be a function with an nth-order derivative at a point and let . Prove that for every non-negative integer.
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 ?
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.
What is a difference between the Maclaurin polynomial of order n and the Taylor polynomial of order n for a function f ?
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