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Problem : Determine if the series

   

converges.

If an 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 an equal the nth 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/rn    

converges for any real number r > 1.

If an is the nth term of the series, then

= =    

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