How do I do this?

Nichelle14 · #1 $Old$ June 24th, 2006, 01:45 PM

Let S1 = sqrt of 6 and Sn+1 = sqrt(6 + Sn) for n>= 1; Prove that Sn converges and find it's limit

I know that I can use the ratio or root test to see if it works.
I believe it is montonic, but not too sure why.

Soroban · #2 $Old$ June 24th, 2006, 05:37 PM

Hello, Nichelle!

We can find the limit with straight Algebra . . .

Quote:

14] Let $S_1 = \sqrt{6}$ and $S_{n+1} = \sqrt{6 + S_n}$ for $n \geq 1$
Prove that $S_n$ converges and find it's limit.

List the first few terms . . .

$S_1 \;= \;\sqrt{6}$

$S_2 \;= \;\sqrt{6 + \sqrt{6}}$

$S_3 \;= \;\sqrt{6 + \sqrt{6 + \sqrt{6}}}$
. $\vdots$ . . . . . . . . . $\vdots$
$S \;= \;\sqrt{6 + \underbrace{\sqrt{6 + \sqrt{6 + \sqrt{6 + \hdots}}}}}$
. . . . . . . . . . . . . . . This is $S$

So we have: . $S \;= \;\sqrt{6 + S}$

Square both sides: . $S^2 \;= \;6 + S$

We have a quadratic: . $S^2 - S - 6 \;= \;0$

. . which factors: . $(S - 3)(S + 2) \;= \;0$

. . and has the positive root: . $S \,= \,3$

Therefore: . $\lim_{n\to\infty} S_n \;=\;3$ . . . obviously, the sequence conveges.

Edit: Sorry ... I had a very stupid typo ... corrected now.

Nichelle14 · #3 $Old$ June 24th, 2006, 06:51 PM

How do you get 2 to be the postivie root?

You have (s-3)(s+2) = 0

so isn't this true
s-3 = 0 which is s = 3

and

s+2 = 0 which is s = -2

Just checking?

Rebesques · #4 $Old$ July 4th, 2006, 10:39 AM

Soroban has got the method right! We just need some extra justification (that boring stuff pure mathematicians bother people with).

You can prove by induction, that the sequence is increasing and bounded above. By a well known theorem of calculus, the sequence converges. This justifies the step
$S \;= \;\sqrt{6 + \underbrace{\sqrt{6 + \sqrt{6 + \sqrt{6 + \hdots}}}}}$

Soroban used to calculate the actual limit

. Now, you can see that all terms of the sequence remain positive; So the limit s=-2 is not an option.

ThePerfectHacker · #5 $Old$ July 4th, 2006, 03:10 PM

Quote:

Originally Posted by Rebesques

(that boring stuff pure mathematicians bother people with).

Thank you Rebesques, Soroban's solution bothered me precisely as you described it.

srulikbd · #6 $Old$ July 5th, 2006, 03:16 AM

can someone please find the formula for the nth number, and then we can just limit n>infinte, right?

Rebesques · #7 $Old$ July 5th, 2006, 05:23 AM

Quote:

can someone please find the formula for the nth number

You mean expressing the sequence as S_n=something without previous S's. That's not always possible, when you need to know the previous terms of the sequence to find the following ones - Such sequences (like the one here) are called recursive.

Maybe we should just be happy we can tackle questions of convergence without seeing a recognizable pattern in the sequence.