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simplyjasper

Member

Registered: 2009-11-15

Posts: 24

Calculus Questions

Just came back from the Calculus exam and I obviously didn't fair well

Some questions that I had difficulties:

(1) Prove that

(2) Determine whether the following series is absolutely convergent, conditionally convergent or divergent

(3) The function

is defined by

    if   

   and   

or

   if   

Prove that the domain of

 
Is

  continuous on

Justify your answer.

Last edited by simplyjasper (2009-11-18 03:02:24)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Calculus Questions

Hi;

Sorry to hear that. Let's look at the first one:

There is no free n so the limit is meaningless. You are left with:

Last edited by bobbym (2009-11-18 12:46:12)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Calculus Questions

Hi simplyjasper;

That is the MaClaurin series for Arctan(x) when you plug x=1 into it.

So it is easy to sum:

Arctan(1) = π / 4

So:

Last edited by bobbym (2009-11-18 12:45:43)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

simplyjasper

Member

Registered: 2009-11-15

Posts: 24

Re: Calculus Questions

Hi simplyjasper;

That is the MaClaurin series for Arctan(x) when you plug x=1 into it.

So it is easy to sum:

Arctan(1) = π / 4

So:

Hi Bobby,

Yes this is a well known series that is equals to

However, it's a 10 marks questions, I think he's looking into how did this formula came about...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Calculus Questions

Hi simplyjasper;

I am not making my point clear.

Shouldn't be no more difficult than differentiation. That is how you get the Maclaurin series for Arctan(x). So if you can form a maclaurin series and you know that Arctan(1) = pi/4 you just put x=1 an you have that series. You have summed it.

This is the series you were given to sum:

This is the maclaurin series for Arctan(x):

Now just put 1 into x and the 2 series are the equal.

Since you know that Arctan(1) = pi / 4 you are done.

Last edited by bobbym (2009-11-18 13:16:16)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

simplyjasper

Member

Registered: 2009-11-15

Posts: 24

Re: Calculus Questions

Hi simplyjasper;

I am not making my point clear.

Shouldn't be no more difficult than differentiation. That is how you get the Maclaurin series for Arctan(x). So if you can form a maclaurin series and you know that Arctan(1) = pi/4 you just put x=1 an you have that series. You have summed it.

This is the series you were given to sum:

This is the maclaurin series for Arctan(x):

Now just put 1 into x and the 2 series are the equal.

Since you know that Arctan(1) = pi / 4 you are done.

Guess that's the only way... I couldn't find anything on the website that prove why is the series going towards to the value except for the fact that when u use 5 values of it u get it corrected to 2 decimal places and if u take 7 u get 3 decimals place and so on...

I'm totally not sure what he wants... I just wrote the series = tan-1(1) = pi/4...
Hopefully i will get some marks for that...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Calculus Questions

Hi simplyjasper;

That is what I would have done. That is definitely the way to sum that series, if he doesn't give you full credit than it isn't your fault.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

simplyjasper

Member

Registered: 2009-11-15

Posts: 24

Re: Calculus Questions

Hi simplyjasper;

That is what I would have done. That is definitely the way to sum that series, if he doesn't give you full credit than it isn't your fault.

Yeah I think my professor is a little bit psycho... He enjoys seeing us fail his paper ...
Any answers for (2) and (3)?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Calculus Questions

Hi;

(2) is divergent. But I am having a little trouble proving that.

Last edited by bobbym (2009-11-21 00:13:42)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

simplyjasper

Member

Registered: 2009-11-15

Posts: 24

Re: Calculus Questions

Hi;

(2) is divergent. But I am having a little trouble proving that.

I was thinking it divergent too...
It's one of the four unseen questions in my paper which consists of more than 30% of our grades =/...

He kinda made the class adopt a phobia in Mathematics