Urgend Geometric series question

In summary, the conversation discusses how to prove the identity 1/(1+x^2) = 1-x^2 + x^4 + (-1)^n*(x^2n-2) + (-1)^n * (x^2n)/(1+x^2) using the geometric series formula and expanding the arctan function. The conversation also covers finding the general term of the series and the sum of the series, eventually arriving at the result of 1/(1-x^2). The conversation includes a few mistakes and corrections, but ultimately leads to the desired proof.
  • #36
First of all, [itex]\sum_{j=0}^{\infty}(-1)^{j}x^{2j} \neq x^{2} + x^{4} + x^{6} + x^{8}[/itex], it is [itex]\sum_{j=0}^{\infty}(-1)^{j}x^{2j}=1-x^{2} + x^{4} - x^{6} + x^{8}...[/itex].
 
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  • #37
No. You MUST learn to read properly and stop writing sloppy and nonsensical maths.

We have:
[tex](-1)^{n}x^{2n}\sum_{j=0}^{\infty}(-1)^{j}x^{2j}=\frac{(-1)^{n}x^{2n}}{1+x^{2}}[/tex]
 
Last edited:
  • #38
And second; you know also this: [itex]\frac{1}{1-q}=1 + q + q^2 + q^3 +...[/itex].

Can you see the connection now?

I strongly suggest you to over the entire problem again and - as arildno is saying - read it properly.
 
  • #39
arildno said:
No. You MUST learn to read properly and stop writing sloppy and nonsensical maths.

We have:
[tex](-1)^{n}x^{2n}\sum_{j=0}^{\infty}=\frac{(-1)^{n}x^{2n}}{1+x^{2}}[/tex]
Then
[tex]\sum_{j=0}^{\infty}(-1)^{j}x^{2j}=1-x^{2} + x^{4} - x^{6} + x^{8} +\cdots + (-1)^{n}x^{2n} = \sum_{j=0}^{\infty}\frac{(-1)^{n}x^{2n}}{1+x^{2}}[/tex]
 
  • #40
arildno said:
We have:
[tex](-1)^{n}x^{2n}\sum_{j=0}^{\infty}=\frac{(-1)^{n}x^{2n}}{1+x^{2}}[/tex]

Sorry arildno, but that doesn't make sense.
 
  • #41
assyrian_77 said:
Sorry arildno, but that doesn't make sense.
You're right, it doesn't I'll fix it right away.
 
<h2>1. What is an urgent geometric series?</h2><p>An urgent geometric series is a type of mathematical series in which each term is multiplied by a constant ratio in order to obtain the next term. The term "urgent" refers to the fact that the series must be solved quickly or immediately due to its importance or relevance in a particular problem or situation.</p><h2>2. How do you find the sum of an urgent geometric series?</h2><p>To find the sum of an urgent geometric series, you can use the formula S = a(1-r^n)/(1-r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. Alternatively, you can also use the formula S = a(1-r)/(1-r), where a is the first term and r is the common ratio, if the series has an infinite number of terms.</p><h2>3. What is the common ratio in an urgent geometric series?</h2><p>The common ratio in an urgent geometric series is the constant number that is multiplied to each term in order to obtain the next term. It is denoted by the letter "r" and is usually a decimal or fraction. For example, in the series 2, 4, 8, 16, 32, the common ratio is 2 because each term is multiplied by 2 to get the next term.</p><h2>4. How can an urgent geometric series be applied in real life?</h2><p>An urgent geometric series can be applied in various real-life scenarios, such as compound interest calculations, population growth, and radioactive decay. It is also commonly used in business and finance, particularly in analyzing the growth or decline of investments or assets over time.</p><h2>5. What is the difference between a finite and infinite urgent geometric series?</h2><p>A finite urgent geometric series has a specific number of terms, while an infinite urgent geometric series has an unlimited number of terms. This means that the sum of a finite series can be calculated using a specific formula, whereas the sum of an infinite series can only be approximated using a formula or through other mathematical methods.</p>

1. What is an urgent geometric series?

An urgent geometric series is a type of mathematical series in which each term is multiplied by a constant ratio in order to obtain the next term. The term "urgent" refers to the fact that the series must be solved quickly or immediately due to its importance or relevance in a particular problem or situation.

2. How do you find the sum of an urgent geometric series?

To find the sum of an urgent geometric series, you can use the formula S = a(1-r^n)/(1-r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. Alternatively, you can also use the formula S = a(1-r)/(1-r), where a is the first term and r is the common ratio, if the series has an infinite number of terms.

3. What is the common ratio in an urgent geometric series?

The common ratio in an urgent geometric series is the constant number that is multiplied to each term in order to obtain the next term. It is denoted by the letter "r" and is usually a decimal or fraction. For example, in the series 2, 4, 8, 16, 32, the common ratio is 2 because each term is multiplied by 2 to get the next term.

4. How can an urgent geometric series be applied in real life?

An urgent geometric series can be applied in various real-life scenarios, such as compound interest calculations, population growth, and radioactive decay. It is also commonly used in business and finance, particularly in analyzing the growth or decline of investments or assets over time.

5. What is the difference between a finite and infinite urgent geometric series?

A finite urgent geometric series has a specific number of terms, while an infinite urgent geometric series has an unlimited number of terms. This means that the sum of a finite series can be calculated using a specific formula, whereas the sum of an infinite series can only be approximated using a formula or through other mathematical methods.

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