Problem : Does the series 1 + (- 1) + 1 + (- 1) + ... converge?

No. The partial sums alternate between 1 and 0 as n→∞.

Problem : Show that 1/n2 converges by grouping terms 2 and 3, 4 through 7, 8 through 15, and so on using the fact that 1/2n converges.

Note that


+ < +  
+ ... + <4  
+ ... + <8  

and so on, so (naively) the series is "less than" the series

1 + 2 +4 +8 + ... = 1 + + + + ...    

which converges.