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.
We can find the limit with straight Algebra . . .
List the first few terms . . .14] Let and for
Prove that converges and find it's limit.
. . . . . . . . . .
. . . . . . . . . . . . . . . This is
So we have: .
Square both sides: .
We have a quadratic: .
. . which factors: .
. . and has the positive root: .
Therefore: . . . . obviously, the sequence conveges.
Edit: Sorry ... I had a very stupid typo ... corrected now.
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
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.
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.can someone please find the formula for the nth number
Maybe we should just be happy we can tackle questions of convergence without seeing a recognizable pattern in the sequence.