How do you determine the convergence or divergence of #Sigma ((-1)^(n+1))/(2n-1)# from #[1,oo)#?
Alternating series, which alternate between having positive and negative terms, often come in the forms
Leibniz's rule, or the alternating series test, can be used to determine if one of these series converges or not.
For an alternating series such as
#lim_(nrarroo)a_n=0" "" "#(the sequence approaches #0#) #a_n>=a_(n+1)" "" "#(the sequence is decreasing)
We see that
We can also show that
And then by cross-multiplying and solving this inequality, which is tedious, we can show that this is always true (at least for
Anyway, we've showed that
Arranging and adding successive terms
and we have