# Who likes a good calculus problem?

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#### blackbrandt

##### That Darn College Student
Hello people!

I'd like to propose a challenge. Find all values of x so the following is true:

I've managed to figure out the following:

The formula for the series is:
(EDIT: Should be (2n)! in the denominator)

And also, I've proven that it's possible for an answer to exist.

Thoughts?

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#### TangoJuliet

##### Well-Known Member
Ooh, Calculus, the one form of higher math I did not take in High School. Trig kicked my butt, though I still passed. I didn't need the credit for Calc, so I dropped it second semester of my Senior year (1989).

#### blackbrandt

##### That Darn College Student
Calc is actually kinda fun. It gets annoying at some points though... Above problem shows my point.

#### timbucktoo

##### Well-Known Member
Staff member
TRF Supporter
Global Mod
Yea fun - I had 4 years of it in college and never used it a day in my life since!

#### Incongruent

##### Well-Known Member
Yea fun - I had 4 years of it in college and never used it a day in my life since!
But did you get a pretty poster to hang on the wall saying you have such and such degree?

#### neil_w

##### Doldrum dweller
TRF Supporter
Hello people!

I'd like to propose a challenge. Find all values of x so the following is true:
I found high school calculus to be quite enjoyable, but the fun started to drain out of it when we did multivariable calculus in college, and then linear algebra convinced me that I didn't actually like math after all.

Anyway, I'm super rusty on this stuff and wouldn't pretend to know how to solve this rigorously, but the way I'd initially attack it would be to note that for the sum to be zero, there must be some negative terms. Therefore, x must be negative, and therefore all the terms with negative exponent are negative. Then I'd separate the positive and negative terms, and their sum is equal to each other. And then... we'll I'm not sure.:wink: I don't have time at the moment to sit down and play with it on paper.

##### Roger Smith
No Calculus used, but ....

[Spoiler ....]

If the first value in the series is 1 (as stated), then x is equal to 1. Just plug zero into the formula for N (either the one above or the correct one below) and solve for x. You'll get 1.

But, if x is 1, the sum of the series won't be 0.

So, my answer is ..... {}

-- Roger

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#### dmorey

##### Well-Known Member
The denimominator for your formula is incorrect. (N+1)! Should be (2N)!

#### Zebedee

##### Well-Known Member
Are we doing your Math homework again?

#### blackbrandt

##### That Darn College Student
The denimominator for your formula is incorrect. (N+1)! Should be (2N)!
Oh whoops! Didn't catch that. Thank you!

#### blackbrandt

##### That Darn College Student
! think the second formula is supposed to be 1 + the sum of the series for n=1 to infinity.

-- Roger
I thought about solving it that way, but then I realized that if I kept it as n=1, I could use it as a power series. The power series for cos(z) is VERY close, and it turns into the original series with 1 substitution (cos(i*sqrt(x))).

So I'm currently dealing with a hyperbolic cosine trying to get my answer

##### Roger Smith
I thought about solving it that way, but then I realized that if I kept it as n=1, I could use it as a power series.
I think you missed my point. If the (corrected) second formula is correct, then there can be no answer.

In the first formula, you've explicitly defined the first value in the series as 1. Solving for x using the second formula (when n = 0), you get x = 1. But, that makes the "= 0" part of both formulas false.

-- Roger

#### Bat-mite

##### Rocketeer in MD
Man, I took Calc in high school, 1983-84, got an A, but did not do advanced placement. So I took it again in college for credit, got an A both semesters (1984-85).

Then I completely forgot it and never used. I toyed with getting an MS in Systems Engineering, so I took a Calc refresher, no grade, but I nailed it. That was in 2005.

And now I can't remember a #[email protected]%\$# thing. I know there was something called a derivative and it has something to do with area under a graph, oh yeah, and limits as something approaches zero or infinity. Right?

Use it or lose it, dude!

#### rc dude

##### Well-Known Member
For the second formula, x~-2.467 according to wolfram alpha

#### Zeus-cat

##### Well-Known Member
Yea fun - I had 4 years of it in college and never used it a day in my life since!
The point of courses like calculus isn't just to teach scientists and engineers to do advanced math. Your approach to problem solving should be more than one dimensional, and that is one of the main lessons to be learned from many college courses; especially for those courses outside your area of expertise. You learn different ways of thinking and can apply a different approach to a problem when the one you always use doesn't work in a particular situation. So I daresay that you have certainly used what you learned in calculus class.

##### Roger Smith
The series, based on the second expression (with the corrected denominator) expands to:

1/x + x/2 + x^2/24 + x^3/720 ....

To be equal to zero, x would be approximately -31 (though there are some complex solutions).

-- Roger

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#### emckee

##### Well-Known Member
what plugin am I missing if I can't see the original equations? :-/

##### Roger Smith
what plugin am I missing if I can't see the original equations? :-/
They are just images. They are png files, but were named as jpg which might confuse some browsers.

I'll attach them here for you and anyone else that can't see them.

-- Roger

#### Zeus-cat

##### Well-Known Member
They are just images. They are png files, but were named as jpg which might confuse some browsers.

I'll attach them here for you and anyone else that can't see them.

View attachment 305459View attachment 305460

-- Roger
As posted by several people the second equation is incorrect. Instead of (n+1)! it should be(2n)!

#### boatgeek

##### Well-Known Member
Everything was great for me in math (including linear algebra) until I hit differential equations. That kicked my butt, twice. When I talk to middle and high school students now, I tell them the dirty secret of engineering is that I haven't done math harder than a square root in 20 years. In general, professionals have computers to do the heavy lifting in the math department. You do need the higher math to understand what is going on and so you can get a feeling of what the answer should be so you can identify when the computer is screwing up.

#### Swissyhawk

##### Well-Known Member
The numerator in the second equation is also incorrect. It should be x^n not x^(n-1) because n starts at 0.

#### Maxitout

##### Well-Known Member
I believe the answer is 42. Oh, wait.... what was the question?:wink:

Phil L.

#### soopirV

##### Well-Known Member
Everything was great for me in math (including linear algebra) until I hit differential equations. That kicked my butt, twice. When I talk to middle and high school students now, I tell them the dirty secret of engineering is that I haven't done math harder than a square root in 20 years. In general, professionals have computers to do the heavy lifting in the math department. You do need the higher math to understand what is going on and so you can get a feeling of what the answer should be so you can identify when the computer is screwing up.
Differential Equations is why I work in biology now instead of chemistry...dang, that was a discouraging year. Turned out okay tho!

#### cerving

##### Owner, Eggtimer Rocketry
TRF Supporter
Series work really good with computers... just plug away until the difference is close to zero. They only had midrange/mainframe systems when I was in college (think punch cards and ASR33 teletypes), but once I learned about numerical integration and convergence to n significant digits I figured that was probably all the calc I was ever going to use in real life.

#### blackbrandt

##### That Darn College Student
OK, so after a lot more work... I've managed to get it down to cosh(sqrtx)=0.

#### Dave A

It took some in high school and in college for a mechanical engineering assoc degree in manufacturing.
I took a cross over course, electromechanical systems course, AutoCAD and learned to programm cnc machines.
I went back to school 15 years after high school. Built everything from houses to a nuclear power plant during that time.
Working in the field helped me to be way ahead of other graduates.
Calculus is necessary but never used it that much in engineering. Computers and control modules did the calcs for me.
My mother was an engineer at a nuclear weapons/fuel facility for 34 years. She said the nubies that came in fresh from Ga Tech weren't taught to read a drawing, unbelievable.

#### Incongruent

##### Well-Known Member
The point of courses like calculus isn't just to teach scientists and engineers to do advanced math. Your approach to problem solving should be more than one dimensional, and that is one of the main lessons to be learned from many college courses; especially for those courses outside your area of expertise. You learn different ways of thinking and can apply a different approach to a problem when the one you always use doesn't work in a particular situation. So I daresay that you have certainly used what you learned in calculus class.

The point of Calculus class is not to teach you about calculus, but to teach you on dealing with intense pain, suffering, frustration, and apathy.

#### blackbrandt

##### That Darn College Student
Well, approximately 6 hours of work later, I have determined that there are no real solutions.

The series is equal to the hyperbolic cosine of the square root of x.