Problem : Does the series 1 + (- 1) + 1 + (- 1) + ^{ ... } converge?
No. The partial sums alternate between 1 and 0 as n→∞ .Problem : Show that 1/n ^{2} converges by grouping terms 2 and 3 , 4 through 7 , 8 through 15 , and so on using the fact that 1/2^{n} converges.
Note that
+ | < | + | |
+ ^{ ... } + | < | 4 | |
+ ^{ ... } + | < | 8 |
1 + 2 +4 +8 + ^{ ... } = 1 + + + + ^{ ... } |
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