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Calculus BC: Series

Problems

Geometric Series and the Ratio Test

The Integral Test

Problem : Determine if the series

   

converges.

If a n is the n -th term of the series, then

= =    

For large n , this quotient is very close to 1/2 . Hence, for all but the first few integers n , the quotient will be ≤3/4 . Since we may disregard the first few terms when considering questions of convergence, the ratio test implies that the series converges.

Problem : Suppose x is a fixed real number. Determine whether or not the series

   

converges.

Letting a n equal the n th term of the series, we have

= =    

Since this ratio is clearly less than, say, 1/2 for all but the first few n , the ratio test implies that it converges.

Problem : Show that the series

1/r n    

converges for any real number r > 1 .

If a n is the n th term of the series, then

= =    

Since 1/r < 1 whenever r > 1 , the ratio test implies that the series converges.

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