Finding x such that 1+x^2+2x^3+x^4+2x^5+x^6... Converges

  • Thread starter AdkinsJr
  • Start date
In summary: Oh yes, haha. I see now. I just typed the original series wrong. It is 1+2x+x^2+2x^3...The grouping should be (1+2x)+(x^2+2x^3)...= (1 + x2 + x3 + …) + (x3 + x5 + …) :wink:
  • #1
AdkinsJr
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I'm trying to review some calc, I went through the series and sequence sections pretty rapidly since my courses were all quarter-length.

I want to find x such that the series converges and find the sum.


[tex]1+x^2+2x^3+x^4+2x^5+x^6...[/tex]

[tex]=(1+x^2)+(2x^3+x^4)+(2x^5+x^6)...(2x^{2n+1}+x^{2n})[/tex]

[tex]\Sigma_{n=0}^{\infty}\left(x^{2n}+2x^{2n+1}\right)=\Sigma_{n=0}^{\infty}\left{(x^2)^n\right+\Sigma_{n=0}^{\infty}2x(x^2)^n[/tex]

So I think that the series converges for

[tex]0< \mid x^2\mid < 1 \rightarrow 0<x<1 [/tex]

and the sum is

[tex]\frac{1}{1-x^2}+\frac{2x}{1-x^2}=\frac{1+2x}{1-x^2}[/tex]

Is this correct? I don't have solution for this one... I'm not comfortable with it.
 
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  • #2
It looks almost right. In your original series there is no 2x term, but later you include it. Also the convergence is for -1 < x < 1.
 
  • #3
Hmm..I'd rather rewrite this as:
[tex]2\sum_{n=0}^{\infty}x^{n}-\sum_{n}^{\infty}(x^{2})^{n}=\frac{2}{1-x}-\frac{1}{1-x^{2}}=\frac{2+2x}{1-x^{2}}-\frac{1}{1-x^{2}}=\frac{1+2x}{1-x^{2}}[/tex]
 
Last edited:
  • #4
mathman said:
It looks almost right. In your original series there is no 2x term, but later you include it.

The term was in the original series, I htink you mean the 2x factor was on the term. That factor came from the following algebra:

[tex]2x^{2n+1}=2xx^{2n}=2x(x^2)^n[/tex]

Edit: Latex isn't working right, I typed 2xx^{2n} on the second step above...

Also the convergence is for -1 < x < 1.

That makes sense...
 
  • #5
If you look at your original series there is no term there that has 2x (just 2x nothing else) in it. But if you look at your sum and put n = 0, the outcome would be 1 + 2x ... so there is a slight flaw, but easely fixed by subtracting 2x from the sigma notation.
 
  • #6
AdkinsJr said:
The term was in the original series, I htink you mean the 2x factor was on the term. That factor came from the following algebra:

[tex]2x^{2n+1}=2xx^{2n}=2x(x^2)^n[/tex]

Edit: Latex isn't working right, I typed 2xx^{2n} on the second step above...



That makes sense...

The 2x I was referring to was simply the missing term (as ojs pointed out).
 
  • #7
mathman said:
The 2x I was referring to was simply the missing term (as ojs pointed out).

Oh yes, haha. I see now. I just typed the original series wrong. It is [tex]1+2x+x^2+2x^3[/tex]...The grouping should be [tex](1+2x)+(x^2+2x^3)...[/tex]
 
  • #8
= (1 + x2 + x3 + …) + (x3 + x5 + …) :wink:
 

1. What is the meaning of "converges" in this equation?

In this equation, "converges" means that the sum of the infinite series, or the value of x, approaches a specific number as we continue to add more terms. It indicates that the series has a finite limit.

2. How can we determine if this equation converges?

We can use a variety of methods to determine if this equation converges. One method is the ratio test, where we take the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, then the series converges. Another method is the root test, where we take the limit of the nth root of the absolute value of the terms. If this limit is less than 1, then the series converges.

3. What is the significance of finding x such that this equation converges?

Finding x such that this equation converges is important because it allows us to find a specific value or range of values for x that satisfies the equation and results in a convergent series. It also helps us understand the behavior and properties of the series.

4. Are there any restrictions on the values of x that can make this equation converge?

Yes, there are restrictions on the values of x that can make this equation converge. For example, if the ratio or root test results in a limit of 1, then the series may converge or diverge depending on the specific values of x. Additionally, some values of x may result in a divergent series.

5. How can we use this information in real-world applications?

This information can be useful in various fields such as physics, engineering, and economics. In physics, it can help us understand the behavior of systems that involve infinite series, such as oscillating systems. In engineering, it can help us analyze and design structures that rely on infinite series, such as bridges and buildings. In economics, it can help us model and predict the growth or decline of investments or markets.

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