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# Prove that the recursive sequence 1 / xn+2 = 1 / xn+1 + 1 / xn converges

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,

# Prove that the recursive sequence xn+1 = (1+xn)1/2 converges

Prove that the sequence whose terms are defined recursively by

converges, and compute the limit of the sequence.

Proof. To show the sequence converges we show that it is monotonically increasing and bounded above. To see that it is monotonically increasing we use induction to prove that

For the case we have

Since , the statement holds in the case . Assume then that the statement holds for some positive integer . Then,

since by the induction hypothesis. Hence, so by induction for all positive integers . Hence, the sequence is monotonically increasing.
Next we use induction again to prove the sequence is bounded above by . For we have so the hypothesis holds. Assume then that for all positive integers up to . Then,

Hence, for all positive integers .
This shows that the sequence converges

To compute the limit, assume the sequence converges to a number (we just proved that it converges, so this assumption is valid). Then we have

(We can discard the negative solution since to the quadratic at the end since the sequence is certainly all positive terms.)

# Prove a formula for the integral from 0 to x of tm(1+t)n

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,

# Establish a recurrence relation for a given integral

Define

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