Solving For x in a Series: sum[n=1..infinity] 4x^(3n) = 42

[kyle1]

I have been tasking with solving for x in the series given at the bottom of this thread. I have no idea where to even start in this case. Any help or hints are appreciated ...

sum[n=1..infinity] 4x^(3n) = 42

 

[mmm4444bot]

I have been [tasked] with solving for x in the series given …

∑n=1∞ 4x3n = 42

d

Is that symbol part of the exercise?

I'm fairly certain that the blue part means:

\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\)

Please clarify the meaning of 4x3n .

(If it's 4x³ⁿ, then there is one Real solution for x -- taking the form of a Rational number times the cube root of an Integer.)


:idea: In the Read Before Posting announcement, there's a link titled, "Formatting Math as Text". You can use it to learn how to type math expressions with a keyboard.

EG:

sum[n=1..infinity] 4x^(3n) = 42

You can also describe stuff in words: "The sum, as n goes from 1 to infinity, of 4*x^(3n) equals 42."

 

[kyle1]

Yea I do not know how the formatting got messed up, I copied it from somewhere it looked fine in the editor. But, you were correct. Its
sum[n=1..infinity] 4x^(3n) = 42. I will edit the original post.

I am still a little confused about your answer, does that then mean I should solve it like
4x^3=42 ignoring the series notation part?

 

[Deleted member 4993]

I have been tasking with solving for x in the series given at the bottom of this thread. I have no idea where to even start in this case. Any help or hints are appreciated ...

sum[n=1..infinity] 4x^(3n) = 42

That means that first few terms of your series would look like:

4x^3 + 4x^6 + 4x^9 + 4x^12 ..... = 42

or

x^3 + x^6 + x^9 + x^12....... = 10.5

Left hand side is a geometric series. Can you express its sum in terms of x^3?

 

[mmm4444bot]

… copied it from somewhere it looked fine in the editor …

Please use the Preview Post button, to check your typing before submitting a post.

I am still a little confused about your answer, does that then mean I should solve it like
4x^3=42 ignoring the series notation part?

Oh, no -- don't ignore the summation! I didn't mention anything about how to begin because I didn't yet know what your typing meant.

I just made a guess regarding the expression 4x3n, and then I confirmed that, if 4x3n means 4x^(3n), then it would be possible to find a Real solution for x. Further, that value of x (in exact form) is the product of a Rational number times the cube root of an Integer.

 

[kyle1]

Thank you for your replies, but I am still pretty lost.
I am not seeing why solving for x in the following doesn't work.
42 = (4x^3)/(1-x)

Is there a step one in solving this problem? I have tried many things and I always come to the incorrect answer.

 

[mmm4444bot]

I am not seeing why solving for x in the following doesn't work.

42 = (4x^3)/(1-x)

That's not working because x is not the common ratio (r).

In other words, you need to substitute the correct expression for r, in the formula for the sum of an infinite geometric sequence (with n starting at 1).

sum = a₁/(1 - r)

Can you answer Subhotosh's question?

What is your definition for the geometric sequence; that is, what are the expressions for a₁ and r, in the formula for the terms (with n starting at 1):

a_n = a₁ * r^(n - 1)

You need to substitute the same expressions for a₁ and r into the formula for the sum. :cool: