Trying to find the quotient of infinite sums

In summary, the conversation discusses how to express the given expression in terms of a rational function and the struggle to do so. It is determined that the denominator should be \frac{1}{1-x} and in order for the expression to work out, the numerator should be \frac{1}{(1-x)(x-1)}. The issue is figuring out how to express 0+x+2x^2+3x^3+... as a function of x, with attempts at integrating and differentiating the series. The suggested solution is to factor out x and then integrate.
  • #1
demonelite123
219
0
i am trying to re-express the following in terms of a rational function: [itex]\frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...} [/itex]. i know that this is supposed to be [itex]\frac{1}{x-1}[/itex] but I can't figure out how to do it.

I know the denominator is just [itex] \frac{1}{1-x}[/itex]. so in order for this work out, the infinite sum which makes up the numerator should be [itex]\frac{1}{(1-x)(x-1)}[/itex]. so my problem is figuring out how to express [itex]0+x+2x^2+3x^3+...[/itex] as a function of x. I have tried integrating/differentiating the series which didn't work and i haven't been able to figure out another way to do this.

can someone help me figure this out?
 
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  • #2
Factor x out first, then integrate.
 
  • #3
oh wow I am really getting rusty on my calculus. thanks for your reply!
 

Related to Trying to find the quotient of infinite sums

What is the concept of finding the quotient of infinite sums?

The concept of finding the quotient of infinite sums is a mathematical process that involves adding an infinite number of terms in a series and determining the resulting value.

Can infinite sums have a quotient?

Yes, infinite sums can have a quotient. When the sum of the terms in a series is finite, the quotient can be calculated as the limit of the sum as the number of terms approaches infinity.

How do you find the quotient of infinite sums?

To find the quotient of infinite sums, you must first determine if the series is convergent or divergent. If the series is convergent, you can use the formula for the sum of an infinite geometric series to find the quotient. If the series is divergent, you can use other mathematical techniques such as integration or partial sums to approximate the quotient.

Are there any practical applications for finding the quotient of infinite sums?

Yes, there are many practical applications for finding the quotient of infinite sums. For example, in physics and engineering, infinite sums are used to model natural phenomena such as sound waves and electrical circuits. In economics and finance, infinite sums are used to calculate the present value of future cash flows.

Is finding the quotient of infinite sums relevant in modern scientific research?

Yes, finding the quotient of infinite sums is still relevant in modern scientific research. It is a fundamental concept in mathematics and is used in various fields such as physics, engineering, and economics. In fact, new developments in mathematics, such as the theory of infinite series, continue to expand our understanding of this concept and its applications.

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