How do you determine whether the sequence #a_n=n(-1)^n# converges, if so how do you find the limit?
1 Answer
The sequence diverges.
Explanation:
We can apply the ratio test for sequences:
Suppose that;
# L=lim_(n rarr oo) |a_(n+1)/a_n| < 1 => lim_(n rarr oo) a_n = 0#
i.e. if the absolute value of the ratio of successive terms in a sequence
So for our sequence;
# a_n = n(-1)^n #
So our test limit is:
# L = lim_(n rarr oo) | ( (n+1)(-1)^(n+1) ) / ( n(-1)^n ) | #
# \ \ \ = lim_(n rarr oo) | ( (n+1)(-1)^n(-1) ) / ( n(-1)^n ) | #
# \ \ \ = lim_(n rarr oo) | ( (n+1)(-1) ) / ( n ) | #
# \ \ \ = lim_(n rarr oo) | ( (n+1) ) / ( n ) | #
# \ \ \ = lim_(n rarr oo) | 1+1/n | #
# \ \ \ > 1 #
And so the sequence does not converge.