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

1 Answer
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.