Nth partial sum of a series Σ Ak is Sn = (n-1)/(n+1)

In summary: I will edit that. The first was correct (A(k) converges if the sequence of partial sums converges).In summary, to find A_k, one must subtract S(n) from S(n+1) and take the limit as n goes to infinity. This method also determines whether the series Σ A_k converges or not. In this case, the limit of S(n) converges to 1, indicating that Σ A_k does indeed converge. There was also a misunderstanding of the question at first, but HallsOfIvy clarified and provided the correct solution.
  • #1
Kenshin
5
0
If the nth partial sum of a series Σ Ak is Sn = (n-1)/(n+1) , find Ak . Does Σ Ak converge?

i looked in my math book and can't find how to do this.
 
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  • #2
Try subtracting S(n) from S(n+1).
 
  • #3
Either I or Tide misunderstands your question. Subtracting Sn from Sn+1 will give the An+1 but won't tell you whether or not the sequence converges.

The DEFINITION of convergence of a series is the convergence of the sequence of partial sums.

Can you find the limit of the sequence [itex]\frac{n-1}{n+1}[/itex] as n goes to infinity??
 
  • #4
Ak= 2/x^2+3x+2. so you subtrack Sn+1-Sn. can u show me how that works. it just seems weird that, that is all u have to do. thanks
 
  • #5
Kenshin,
Look at the last statement made by HallsOfIvy ...
Have u worked out what he said?

-- AI
 
  • #6
Kenshin said:
Ak= 2/x^2+3x+2. so you subtrack Sn+1-Sn. can u show me how that works. it just seems weird that, that is all u have to do. thanks

To find [tex]A_k[/tex] yes, this is all you have to do (you replaced your "k" with "x" though, and also "n"). Do you understand why this works?

To answer the question of convergence for the series, see HallsofIvy's post. You have to look at [tex]\lim_{n\rightarrow\infty}S_n[/tex]
 
  • #7
Kenshin said:
Ak= 2/x^2+3x+2. so you subtrack Sn+1-Sn. can u show me how that works. it just seems weird that, that is all u have to do. thanks


Let A(k) be any series, and S(n) the sum of the first n terms, then S(n+1) is the sum of the first n terms plus the n+1'st term. Can you now prove the result that puzzles you?
 
  • #8
I DID misunderstand the question! There were TWO things asked:

1) Find A(k) which is what Tide was responding to.

Since S(n), the "partial sum is defined as A(1)+ A(2)+ ...+ A(n),
A(n)= S(n)- S(n-1)= (n-1)/(n+1) - (n-1-1)/(n-1+1)= (n-1)/(n+1)- (n-2)/n
= n(n-1)/n(n+1)- (n+1)(n-2)/n(n+1)= (n2-n-n2+n+2)/n(n+1)
= 2/n(n+1).

2) Since A(k) converges if and only if the sequence of partial sums converges, look at
S(n)= (n-1)/(n+1). Divide both numerator and denominator by n: (1- 1/n)/(1+ 1/n).
As n goes to infinity, 1/n goes to 0 so the S(n) converges to 1.

Yes, Σ A(k) converges. In fact, it converges to 1.
 
  • #9
I don't think you mean if and only if in part 2) there.
 
  • #10
matt grime said:
I don't think you mean if and only if in part 2) there.


? That's certainly the definition of "convergence of a series" that I learned.

If the sequence of partial sums converges, then the series converges to the same limit.

If the sequence of partial sums does not converge, then the series does not converge.
 
  • #11
HallsofIvy said:
2) Since Σ A(k) converges if and only if the sequence of partial sums converges,

I inserted missing Σ?
 
  • #12
The reason I didn't put a correction is that two spring to mind. A(k) converges if the sequence of partial sums converges, or by definition the sum of the sequence converges if and only if the sequence of partial sums converges, and I wasn't sure which Hallsofivy meant. It seems from your reply, that you intended the second option.
 
  • #13
Oh- I forgot the Σ!
 

FAQ: Nth partial sum of a series Σ Ak is Sn = (n-1)/(n+1)

What is the formula for the Nth partial sum of a series?

The formula for the Nth partial sum of a series is Sn = (n-1)/(n+1), where n represents the number of terms in the series.

How is the Nth partial sum of a series calculated?

The Nth partial sum of a series is calculated by multiplying the number of terms (n) by the value of the series at that term, and then adding all of these values together.

What does the formula (n-1)/(n+1) represent in the context of a series?

The formula (n-1)/(n+1) represents the sum of the first n terms in a series, where n is the number of terms in the series. It is also known as the Nth partial sum of the series.

How is the Nth partial sum of a series useful in mathematical analysis?

The Nth partial sum of a series is useful in mathematical analysis as it allows us to approximate the value of an infinite series by considering only a finite number of terms. This can help us understand the behavior and convergence of a series.

Can the Nth partial sum of a series be used to find the exact value of an infinite series?

No, the Nth partial sum of a series can only provide an approximation of the value of an infinite series. To find the exact value, we would need to consider an infinite number of terms, which is not possible in most cases.

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