Example: The series does not converge.

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Consider the sequence of partial sums:

• S _n = -1 + 1 - 1 + 1 ... - 1 (if n is odd)
• S _n = -1 + 1 - 1 + 1 ... - 1 + 1 (if n is even)

But then we have that

• S _n = -1 if n is odd and 0 if n is even

or, in other words, the sequence of partial sums is basically the same as the sequence

• { (-1) ⁿ }

This sequence diverges, as proved before. Hence, our sequence of partial sums - while bounded - does not converge as well, and therefore the series is divergent.

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(bgw)