Prove that the sequence be defined by the recursive relationship,
converges and find the limit of the sequence.
Proof. First, we show that the sequence is monotonically decreasing for all . For the base case we have and
Hence, . Assume then that for all positive integers up to some we have . Then,
Thus, the sequences is monotonically decreasing. The sequence is certainly bounded below since all of the terms are greater than 0. Therefore, the sequence converges
To compute the limit of the sequence, assume the sequence converges to a finite limit (justified since we just proved that it does indeed converge). Therefore,
Define a function
Prove that the following relation holds,
Using this find the value of .
First, we evaluate by parts, letting
This gives us
Multiplying both sides by we have
Then, to compute we first compute to use in our recurrence,
Then, applying the recurrence,
Prove that this integral satisfies the recurrence relation,
Using this compute , and .
Proof. We start by integrating by parts with
Therefore we have
Now, to compute , and using the recurrence relation we first evaluate to get ourselves started,
Applying the recurrence relation with we have
Then, applying the recurrence relation with (and ) we have
Next, applying the recurrence relation with (and ) we have
Finally, applying the recurrence relation with (and ) we have