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Calculus BC: Series

Problems

Series With Positive and Negative Terms

Power Series

Problem : Give an example of a series that converges but does not converge absolutely.

Consider the series

1 - + - + ...    

Convergence follows from the alternating series test, whereas absolute convergence fails because the harmonic series diverges.

Problem : Prove that (- 1)n e -n2 converges.

The result follows from the alternating series test by noting that e -(n+1)2 e -n2 and that e -n2 = 0 .

Problem : Determine whether or not

(- 1)n    

converges.

The series converges by the alternating series test, since the absolute value of the n -th term in the series is

   

Notice that the convergence is not absolute.

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