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Interval and radius of convergence

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1. Find the interval of convergence (including a check of end-points) for each of the given power series.

2. Use the geometric series test (GST) to write each of the given functions as a power series centred at x=a, and state for what values of x the series converges.

3. Use known Maclaurin series for e^x, 1/(1-x), and sin(x) to derive Maclaurin series for the given functions. State the operations used, and the radius of convergence of the series derived.

4. In problem 3, the radius of convergence was known from theorems involving the various operations. In this problem, complete the determination of the interval of convergence by checking the end-points of each of the series found in problem 3.

5. Find the specified Taylor polynomial P_N(x) centered at x=a for each of the given functions by evaluating f, f', f'', ... at x=a to determine the coefficients.

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This solution shows all the steps to solve the given questions regarding the interval and radius of convergence.

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