Interactive Real Analysis
- part of Examples 4.1.5(a):
The series 
does
not converge.
Back
Consider the sequence of partial sums:
does
not converge.
BackS n = -1 + 1 - 1 + 1 ... - 1 = -1if n is odd, and
S n = -1 + 1 - 1 + 1 ... - 1 + 1 = 0if n is even.
S n = -1 if n is odd and 0 if n is evenor, in other words, the sequence of partial sums is the same as the sequence
{ 1/2 ((-1)n-1)}This sequence diverges, similar to the sequence {(-1)n} we discussed previously. Hence, our sequence of partial sums - while bounded - does not converge and therefore the series is divergent.