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Calculus BC: Series

Problems

Convergence of Series

The Comparison Test

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 1/2n converges.

Note that


+ < +  
+ ... + < 4  
+ ... + < 8  

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

1 + 2 +4 +8 + ... = 1 + + + + ...    

which converges.

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