Calculus BC: Series

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

Solution for Problem 1 >>

Consider the series

1 - + - + ...

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

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

Solution for Problem 2 >>

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

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Problem : Determine whether or not

(- 1)n

Solution for Problem 3 >>

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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