So one of the prerequisites for applying the alternating series test is that the serie is of the form $$\sum_{n=1}^{\infty} (-1)^{n-1}b_n$$
Where $b_n$ has to be greater than 0. I was wondering why $b_n$ has to be greater than zero?
It would obviously still alternate if $b_n$ was less than zero, so why is this a condition?
Thanks in advance!
Best Answer
There is no principal difference between series where all $b_n > 0$ and where all $b_n< 0$ from convergence point of view. Convergence of 2 such series are equivalent and sum differs, when exists, by $−1$.
Analogically, there is no difference in alternating series from convergence point of view if we consider all $b_n > 0$ and/or where all $b_n< 0$. If $b_n < 0$, then we can write $b_n = - (-b_n)$ and take minus out of summing in partial sums and, if series converged, out of infinite sum and consider convergence question for $(-b_n) > 0$: $\sum\limits_{n=1}^{\infty} (-1)^{n-1}b_n=-\sum\limits_{n=1}^{\infty} (-1)^{n-1}(-b_n)$.