# Really 2 need your help

#### Otis

##### Senior Member

What have you done, so far? Where did you get stuck?

#### MethMath11

##### Junior Member

What have you done, so far? Where did you get stuck?

The absolute X thinggy

#### Otis

##### Senior Member
The absolute X thinggy
That thinggy is called an inequality statement, and the variable symbol is x, not X.

The inequality |x|<1 means -1<x<1, and it informs us that the common ratio in the given geometric series is between -1 and 1. That is a requirement; otherwise, the given infinite geometric series S would not sum (i.e., converge) to a finite value.

$$\;$$

#### Dr.Peterson

##### Elite Member
What have you learned about infinite series that look like that? How about slightly simpler ones like 1 + x + x^2 + ..., or 1 + x^2 + x^4 + ...? Anything like that may be useful.

Don't worry about the inequality; that could be a clue, but it's not central to the problem. It's just there so that what you have to do will work.

#### Otis

##### Senior Member
@MethMath $$\;$$ For the approach I took, the formula (referenced in post #4) for summing a convergent infinite series is central. In hindsight, I realize the multiple choices infer that S is a real number, so in that regard the author's inclusion of |x|<1 is just good form (for completeness), and thus you may jump directly to the formula you were taught for the sum of an infinite geometric series (used when one knows or assumes the sum must exist).

You might also find helpful the ability to decompose a single ratio into a difference of ratios. I mean, for example, knowing how to rewrite a ratio like x/(1−x) in the form 1/a−1/b. (There are different methods for that.)

Is this exercise from a school assignment?

#### MethMath11

##### Junior Member
@MethMath $$\;$$ For the approach I took, the formula (referenced in post #4) for summing a convergent infinite series is central. In hindsight, I realize the multiple choices infer that S is a real number, so in that regard the author's inclusion of |x|<1 is just good form (for completeness), and thus you may jump directly to the formula you were taught for the sum of an infinite geometric series (used when one knows or assumes the sum must exist).

You might also find helpful the ability to decompose a single ratio into a difference of ratios. I mean, for example, knowing how to rewrite a ratio like x/(1−x) in the form 1/a−1/b. (There are different methods for that.)

Is this exercise from a school assignment?

Yes. That explain alot. But still, the be written 1/p - 1/q part, I thought it has something with the absolute X, and turns out it has nothing to do with it. Now I'm confused.

Sorry for asking, but could you send me the answer with step by step on how to answer it?

#### Otis

##### Senior Member
… [the] 1/p - 1/q part …
Now I'm confused …
Are you confused about partial fraction decomposition? That is, if you need lessons on how to do it, then you could google those keywords. There are many lessons and examples online.

Sorry for asking again, but is this exercise part of a school assignment?
$\;$

#### MethMath11

##### Junior Member
Are you confused about partial fraction decomposition? That is, if you need lessons on how to do it, then you could google those keywords. There are many lessons and examples online.

Sorry for asking again, but is this exercise part of a school assignment?
$\;$
No, self study for future tests

#### ksdhart2

##### Senior Member
I'm going to be frankly honest with you, and I hope that in doing so I don't come off as rude... if, as your replies in this thread seem to indicate, you're uncertain about what absolute value notation means, how it relates to this problem, or why the given condition is mandatory, I sincerely doubt you're prepared for this problem. In fact, you may not even be prepared for whatever math course you're in. Now, there's no real shame in this - students often get placed in the wrong math course and out of a sense of "politeness" and/or wanting to "tough it out" simply suffer through rather than correcting the error.

Oh, actually, I've just seen the most recent reply in which you say this is a bit of self-study. In that case, the bit about math courses isn't particularly relevant, but my opinion that it seems you're not ready to tackle this type of problem still stands. To get ready for this material, you'll want to review absolute value notation but your primary area of study needs to be geometric series, and then, as Otis said, secondarily study partial fraction decomposition. I've included links where you can find study materials on these two topics. Just as a rough guideline, these are the sub-questions you'll need to be able to answer in order to solve this problem:
• The given series $$S$$ is a geometric series. Can you recognize why this is the case? What is the general term of this geometric series?
• A geometric series with infinitely many terms can only sum to a finite number if a critical condition on the variable (here $$x$$) is met. What is that condition? Can you see why the problem text ensures this condition is satisfied?
• There's a handy formula for what an infinite geometric series sums to. Do you know this formula? This formula has variables $$a$$ and $$r$$. Do you know what they are for this problem? What does that make the sum in question?
• What does it mean for this sum you just found, in terms of $$x$$, to be equal to $$\frac{1}{P} - \frac{1}{Q}$$? (Hint: This is where partial fraction decomposition comes in)
• Finally, what does that allow you to say about $$P + Q$$?

#### MethMath11

##### Junior Member
I'm going to be frankly honest with you, and I hope that in doing so I don't come off as rude... if, as your replies in this thread seem to indicate, you're uncertain about what absolute value notation means, how it relates to this problem, or why the given condition is mandatory, I sincerely doubt you're prepared for this problem. In fact, you may not even be prepared for whatever math course you're in. Now, there's no real shame in this - students often get placed in the wrong math course and out of a sense of "politeness" and/or wanting to "tough it out" simply suffer through rather than correcting the error.

Oh, actually, I've just seen the most recent reply in which you say this is a bit of self-study. In that case, the bit about math courses isn't particularly relevant, but my opinion that it seems you're not ready to tackle this type of problem still stands. To get ready for this material, you'll want to review absolute value notation but your primary area of study needs to be geometric series, and then, as Otis said, secondarily study partial fraction decomposition. I've included links where you can find study materials on these two topics. Just as a rough guideline, these are the sub-questions you'll need to be able to answer in order to solve this problem:
• The given series $$S$$ is a geometric series. Can you recognize why this is the case? What is the general term of this geometric series?
• A geometric series with infinitely many terms can only sum to a finite number if a critical condition on the variable (here $$x$$) is met. What is that condition? Can you see why the problem text ensures this condition is satisfied?
• There's a handy formula for what an infinite geometric series sums to. Do you know this formula? This formula has variables $$a$$ and $$r$$. Do you know what they are for this problem? What does that make the sum in question?
• What does it mean for this sum you just found, in terms of $$x$$, to be equal to $$\frac{1}{P} - \frac{1}{Q}$$? (Hint: This is where partial fraction decomposition comes in)
• Finally, what does that allow you to say about $$P + Q$$?
The problem was the direct translation, a simplefying the question so I can translate it into my native language, I was hoping for an explanation on what's the meaning behind "be written", but thx for the help, really appreciate it

#### MethMath11

##### Junior Member
I'm going to be frankly honest with you, and I hope that in doing so I don't come off as rude... if, as your replies in this thread seem to indicate, you're uncertain about what absolute value notation means, how it relates to this problem, or why the given condition is mandatory, I sincerely doubt you're prepared for this problem. In fact, you may not even be prepared for whatever math course you're in. Now, there's no real shame in this - students often get placed in the wrong math course and out of a sense of "politeness" and/or wanting to "tough it out" simply suffer through rather than correcting the error.

Oh, actually, I've just seen the most recent reply in which you say this is a bit of self-study. In that case, the bit about math courses isn't particularly relevant, but my opinion that it seems you're not ready to tackle this type of problem still stands. To get ready for this material, you'll want to review absolute value notation but your primary area of study needs to be geometric series, and then, as Otis said, secondarily study partial fraction decomposition. I've included links where you can find study materials on these two topics. Just as a rough guideline, these are the sub-questions you'll need to be able to answer in order to solve this problem:
• The given series $$S$$ is a geometric series. Can you recognize why this is the case? What is the general term of this geometric series?
• A geometric series with infinitely many terms can only sum to a finite number if a critical condition on the variable (here $$x$$) is met. What is that condition? Can you see why the problem text ensures this condition is satisfied?
• There's a handy formula for what an infinite geometric series sums to. Do you know this formula? This formula has variables $$a$$ and $$r$$. Do you know what they are for this problem? What does that make the sum in question?
• What does it mean for this sum you just found, in terms of $$x$$, to be equal to $$\frac{1}{P} - \frac{1}{Q}$$? (Hint: This is where partial fraction decomposition comes in)
• Finally, what does that allow you to say about $$P + Q$$?
And now I'm not confused anymore, really glad that you elaborate everything for me to understand

#### MethMath11

##### Junior Member
I'm going to be frankly honest with you, and I hope that in doing so I don't come off as rude... if, as your replies in this thread seem to indicate, you're uncertain about what absolute value notation means, how it relates to this problem, or why the given condition is mandatory, I sincerely doubt you're prepared for this problem. In fact, you may not even be prepared for whatever math course you're in. Now, there's no real shame in this - students often get placed in the wrong math course and out of a sense of "politeness" and/or wanting to "tough it out" simply suffer through rather than correcting the error.

Oh, actually, I've just seen the most recent reply in which you say this is a bit of self-study. In that case, the bit about math courses isn't particularly relevant, but my opinion that it seems you're not ready to tackle this type of problem still stands. To get ready for this material, you'll want to review absolute value notation but your primary area of study needs to be geometric series, and then, as Otis said, secondarily study partial fraction decomposition. I've included links where you can find study materials on these two topics. Just as a rough guideline, these are the sub-questions you'll need to be able to answer in order to solve this problem:
• The given series $$S$$ is a geometric series. Can you recognize why this is the case? What is the general term of this geometric series?
• A geometric series with infinitely many terms can only sum to a finite number if a critical condition on the variable (here $$x$$) is met. What is that condition? Can you see why the problem text ensures this condition is satisfied?
• There's a handy formula for what an infinite geometric series sums to. Do you know this formula? This formula has variables $$a$$ and $$r$$. Do you know what they are for this problem? What does that make the sum in question?
• What does it mean for this sum you just found, in terms of $$x$$, to be equal to $$\frac{1}{P} - \frac{1}{Q}$$? (Hint: This is where partial fraction decomposition comes in)
• Finally, what does that allow you to say about $$P + Q$$?
So basically
Sinf = 2x/1-x^2 = 1/p - 1/q right?

#### MarkFL

##### Super Moderator
Staff member
So basically
Sinf = 2x/1-x^2 = 1/p - 1/q right?
You would have:

$$\displaystyle \frac{2x}{1-x^2}=\frac{1}{P}-\frac{1}{Q}$$

Now you want to apply partial fraction decomposition to the LHS.

#### MethMath11

##### Junior Member
You would have:

$$\displaystyle \frac{2x}{1-x^2}=\frac{1}{P}-\frac{1}{Q}$$

Now you want to apply partial fraction decomposition to the LHS.
I got 2x = Q- P while 1-X^2 = P.Q
I tried substitute one with the other, didn't get the result

#### MethMath11

##### Junior Member
Here's what I got so far.
P and Q are Coefficient of an polynomial equation, so something like PX^3 + QX^2 + RX + S
Q-P = 2X
And
P.Q = 1- x^2

If there's a mistakes,please do tell

#### Jomo

##### Elite Member
P and Q are Coefficient of an polynomial equation, so something like PX^3 + QX^2 + RX + S
No, from what I see you clearly posted in your original post that P and Q are polynomials.

#### MarkFL

##### Super Moderator
Staff member
I got 2x = Q- P while 1-X^2 = P.Q
I tried substitute one with the other, didn't get the result
You want to use partial fractions so that you get the form:

$$\displaystyle \frac{2x}{1-x^2}=\frac{1}{f(x)}-\frac{1}{g(x)}$$

There are two possibilities, so you have to choose the one that gives you one of the listed options as the answer.

#### Jomo

##### Elite Member
Try factoring 1-x^2.

#### Otis

##### Senior Member
… I was hoping for an explanation on what's the meaning behind "be written" …

They said, "Let S … be written as 1/P−1/Q". That simply means they want you to rewrite S in the form 1/P−1/Q. In other words, what do the polynomials P and Q need to be, so that:

1/P − 1/Q = 2x/(1 − x^2)

Sinf = 2x/(1-x^2) = 1/p - 1/q right?
Yes, but note the grouping symbols added in red. (These are needed, when typing numerators or denominators that contain more than one term.)

Also, please use symbols as given: S, x, P, Q (not Sinf, X, p, q).

MarkFL said:
… apply partial fraction decomposition …
I got 2x = Q- P while 1-X^2 = P.Q …
That's not a correct method for decomposition. Did you google for any lessons or examples, as suggested? (Maybe you're just guessing.)