How do you determine whether the sequence #a_n=n(-1)^n# converges, if so how do you find the limit?

1 Answer
Steve M
Mar 2, 2017

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 #{a_n}# approaches a limit #L#, and if #L < 1#, then the sequence itself converges to #0#. It is important to note that this is a statement about the convergence of the sequence #{a_n}#, and it is not a statement about the series #sum a_n#.

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.

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