So, I need to test the following series for convergence or divergence:
$$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$
I know that when you use the Alternating Series Test, the series must satisfy two conditions. Which are:
- $$b_{n+1} \le b_n $$
- $$\lim_{n\to \infty} b_n =0$$
I having a hard time with the first condition because if I use 1 for n then I have a problem. This is my work so far:
$$ {(n+1)\over 2^{(n+1)}} ? {n\over 2^n}$$ $$ {(1+1)\over 2^{(1+1)}} ? {1\over 2^1}$$ $$ {(2)\over 2^{(2)}} ? {1\over 2^1}$$ $$ {2\over 4} = {1\over 2}$$ They end up equaling each other. On the other hand, if I plug in 2, I get something that does satisfy the first condition.$$ {(n+1)\over 2^{(n+1)}} ? {n\over 2^n}$$ $$ {(2+1)\over 2^{(2+1)}} ? {2\over 2^2}$$ $$ {(3)\over 2^{(3)}} ? {2\over 2^2}$$ $$ {3\over 8} ? {2\over 4}$$ $$ {3\over 8} \le {1\over 2}$$
So... what do I do? Thanks in advance