Question

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

Answer 1

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.