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.