Question

How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given `1-2+3-4+...+n(-1)^(n-1)`?

Answer 1

`sum_(n=1)^oo n(-1)^(n-1)`

is indeterminate.

Explanation:

The series does not converge as it does not satisfy the necessary condition for convergence:

`lim_(n->oo) a_n = 0`

We can evaluate the partial sums noting that for `n` even we can group the terms as:

`s_(2n) = (1-2)+(3-4)+...(2n-1-2n) =underbrace(-1-1+...-1)_(n " times") =-n`

while for `n` odd:

`s_(2n+1) = 1 + (-2 +3) + (-4+5)+...+(-2n+2n+1) = underbrace(1+1+1+...+1)_(n+1 " times") = n+1`

Hence the sums oscillate and are unbounded in absolute value so that:

`lim_(n->oo) s_n`

does not exist.