How do you test the alternating series #Sigma (-1)^n# from n is #[1,oo)# for convergence?

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Answer 1 · 2017-01-30T20:57:42

the series is indeterminate.

We can easily see that the series is not convergent, since:

#lim_(n->oo) (-1)^n != 0#

We can take a closer look at the partial sums:

#sum_(n=1)^oo (-1)^n#

#s_1 = -1#
#s_2 =0#
#s_3 = -1#
#...#

and we can prove by induction that:

#{(s_(2n) = 0),(s_(2n+1) = -1):}#

so that partial sums oscillate between the two values and do not converge to a limit.