Rearranging Series to Equal SQRT2: How to Solve

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  • #51
would it be s = 5/4 /(1-0.8)
= 6.25

correct?
 
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  • #52
Yes, that is OK!
 
  • #53
so it converges to 6.25 as n tends to infinity?
 
  • #54
Yes, it will :smile:
 
  • #55
I have another series I have to describe and describe whether it converges micromass:
Its:

1+Pi/e + Pi^2/e^2+Pi^3/e^3

Could this be written as:

sum (Pi/e)^n

when n = 0

1

when n =1

Pi/e

etc...
 
  • #56
Well, this series is another "special" series isn't it?
 
  • #57
Ye I've established its a geometrics series with ratio of Pi/e. and it diverges as r>1 correct?
 
  • #58
correct!
 
  • #59
Great I do have one final problem I am stuck with. I have to use simple algebra to find the limit of:

Sqrt(n^2 + 3n) - n

I startd by multiplying top and bottom by Sqrt(n^2 + 3n) + n):

Which gives:

(Sqrt(n^2 + 3n) - n).(Sqrt(n^2 + 3n) + n)/Sqrt(n^2 + 3n) + n)

whcoh can be simplified to:

3n/Sqrt(n^2 + 3n) + n)

correct?? if so where do i go from here?
 
  • #60
Factor out n in \sqrt{n^2+3n}+n. Then you can eliminate the n in the numerator...
 
  • #61
so that would give: n SQRT(n + 3) + 1

correct?
 
  • #62
No, that would correspond to

n(\sqrt{n+3}+1)=\sqrt{n^3+3n^2}+n
 
  • #63
Hmm i a little confused as to how u got the above? I mite need little more help factoring n.
 
  • #64
You do know that \sqrt{n^2a}=n\sqrt{a} right? That's all I did...
 
  • #65
Ah i actually didnt know that! so with that in min I obtain:

n SQRT(1+3/n) + 1

Am I almost there??
 
  • #66
Ah, yes, that is correct. So our limit is

\lim_{n\rightarrow +\infty}{\frac{3n}{n(\sqrt{1+3/n}+1)}

Cancel the n. Then substitute n=+\infty, and see what you get...
 
  • #67
Yes i see so would be:

3/SQRT(1+3/n) +1

So as n tends to infinty the 3/n becomes zero leaving:

3/SQRT(1) + 1

which is 3/2 or 1.5
Correct?
 
  • #68
Yes, you'll get 3 in the numerator. But what happens if you substitute infinity in the denumerator?
 
  • #69
Is post 67 correct micromass?
 
  • #70
Yes, 3/2 is the correct answer.
 
  • #71
fantastic I just have to use the binomial therom now it says to prove this. what part would i apply this to?
 
  • #72
Think I have solved it:

Using (1+x)^r where r is not a whole number. I obtain:

1+3/2.(n) - 1/8 - 1/16 ...

so as the expansion continues the values get closer to zero leaving 1 + 3/2(n)
 
  • #73
I realize I may have ae a mistake. Am i on the rite track?
 
  • #74
Jamiey1988 said:
fantastic I just have to use the binomial therom now it says to prove this. what part would i apply this to?

Use the binomial theorem to prove what??
 
  • #75
the limit I think.
 
  • #76
No, I'm just confused. What exactly are you trying to prove now?
 
  • #77
My original question stated. Using simple algebra find the limit of this sequence as n tends to infinity. Then confirm this using the binomial therom.
 
  • #78
I have no idea what they mean with "confirm with the binomial theorem"...

Do they mean this:

(x-y)^r=\sum_{k=0}^{+\infty}{\binom{r}{k}x^{r-k}y^k}

if so, you just need to substitute x=n2, y=3n and r=1/2...
 
  • #79
Possibly, if that is the case I just sub in values for x y and r. Then what?
 
  • #80
Calculate (n^2-n)^{1/2}-n with the binomial theorem. What does that give you?
 
  • #81
Sorry I am nt to sure how to do that when r = 1/2
 
  • #82
Ok well I am going to come back to that. The bext question I am asked is to describe the sequence that SQRT(n^2 +3n) -n generates:

Substituting in values for n I obtain:

0,1,((SQRT10)-2),((SQRT18)-3), ((SQRT28) -4),...

From this am i correct in saying it is positive and monotonic, And converges to 3/2 as already discovered. Are there any upper lower bounds?
 
  • #83
You'll have to prove that it is positive and monotonic...
 
  • #84
Ok well monotonic is where the next value is greater than or equal to the previous one correct?? So can't I just set two values next to each other:

1<((SQRT10) -2)
 
  • #85
Yes, but this only proves it for n=1 and n=2. You'll need to show it for every n. You'll need to show that for every n

\sqrt{n^2+3n}-n\leq \sqrt{(n+1)^2+3(n+1)}-(n+1)
 
  • #86
Ah ok so from what u have written above. As the sequence is increasing we can say:
an + (an+1) is always positive.

so an + (an+1) = SQRT(n^2 +3n)-n + SQRT((n+1)^2 +3(n+1)) -(n+1)
 
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  • #87
No, you need to prove that...
 
  • #88
Scratch that above should it be an+1 - an is always positive??
so I will have:

an+1 - an = + SQRT((n+1)^2 +3(n+1)) -(n+1) - SQRT(n^2 +3n)-n
 
  • #89
Yes.

(note: the forum rules explicitely forbid that you edit posts that allready have been answered to. So please do not do this)
 
  • #90
Rite so is that just the answer then??
 
  • #91
Yes, I suppose so...
 
  • #92
Great Thank u micromass now I just need to work out the binomia therom part of the question.
 
  • #93
Hey micromass could u possibly help me on a thread I've created regarding a partial differential equation. I am having a little trouble. Thanks in advance.
 
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