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Calculus BC: Series

Problems

Power Series

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Problem : What is the radius of convergence of x n ? of (- 1)n x n ?

Since the radius of convergence only has to do with absolute convergence, the answer to the two parts will be the same. Since x n is a geometric series, it converges whenever x≤1 , but does not converge when x = 1 . Therefore, r = 1 is its radius of convergence.

Problem : Using the fact that

e x = 1 + x + + + ... + + ...    

whenever the right hand series makes sense, find a series that converges to e .

We have seen that the radius of convergence of the right-hand series is . We can therefore substitute x = 1 into this equation, and

e = 1 + 1 + + + ... + + ...    

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