Proof: Compare two integral(Please look at my surgested proof)

[Hummingbird25]

This is a repost, the reason I feared that people who miss the the original.

Homework Statement

Looking at the Integral

a_n = \int_{0}^{\pi} \frac{sin(x)}{x+n\pi}

prove that a_n \geq a_{n+1}

Homework Equations

The Attempt at a Solution

Here is my now proof:

the difference between the two integrals, we seek to show:

$\forall n\in\mathbb{N}:\int_0^\pi\left({\sin t\over t+n\pi}-{\sin t\over t+(n+1)\pi}\right)\,dt\ge 0

Common denominator:

$=\int_0^\pi\left({\sin t((t+(n+1)\pi)-(t+n\pi))\over (t+n\pi)(t+(n+1)\pi)}\right)\,dt

$=\pi\int_0^\pi\left({\sin t\over (t^2+(2n+1)\pi t+(n^2+n)\pi)}\right)\,dt

From here we can use the fact that the denominator on the half of the interval where sinus is negative is larger than the denominator of each and every other corresponding point on the other half of the interval, so the whole integral must be positive.

q.e.d.

How does it look now?

Sincerely Maria.

 

[Dick]

But sin(t) isn't negative anywhere on [0,pi]. That makes life a lot easier.

 

[genneth]

Looks good.

 

[Hummingbird25]

Hi Dick,

So I change my conclusion.

Since sin(t) is positive on the interval [0,pi] then the who integral most be positive and thusly convergent.

Does this sound better?

Sincerely

Maria

 

[Dick]

If everything is nonnegative all you need is that t+n*pi<t+(n+1)*pi. So 1/(t+n*pi)>1/(t+(n+1)*pi). It becomes pretty obvious that your integral is positive.

 

[Hummingbird25]

If everything is nonnegative all you need is that t+n*pi<t+(n+1)*pi. So 1/(t+n*pi)>1/(t+(n+1)*pi). It becomes pretty obvious that your integral is positive.

And thusly it converges?

Sincerely
Maria.

 

[Dick]

And thusly it converges?

Sincerely
Maria.

It converges because the integrand and domain of integration are bounded, right?

 

[Hummingbird25]

It converges because the integrand and domain of integration are bounded, right?

Here is my original proof:

Looking at the Integral

a_n = \int_{0}^{\pi} \frac{sin(x)}{x+n\pi}

prove that a_n \geq a_{n+1}

Homework Equations

The Attempt at a Solution

Here is my now proof:

the difference between the two integrals, we seek to show:

$\forall n\in\mathbb{N}:\int_0^\pi\left({\sin t\over t+n\pi}-{\sin t\over t+(n+1)\pi}\right)\,dt\ge 0

Common denominator:

$=\int_0^\pi\left({\sin t((t+(n+1)\pi)-(t+n\pi))\over (t+n\pi)(t+(n+1)\pi)}\right)\,dt

$=\pi\int_0^\pi\left({\sin t\over (t^2+(2n+1)\pi t+(n^2+n)\pi)}\right)\,dt

The integral is therefore non-negativ and therefore

t+n*pi&lt;t+(n+1)*pi and /(t+n*pi)&gt;1/(t+(n+1)*pi). thus the integral is positive.

The integral is therefore bounded since the integrand is continuous which makes the original in-equality true.

q.e.d.

Is it on the mark now?

Thanks in advance

Sincerely Yours
Maria.

 

[Dick]

i) Your 'therefores' are somewhat backwards. sin(t)/(t+n*pi)>sin(t)/(t+(n+1)*pi) implies the integrand is positive. The integrand being positive DOESN'T imply the inequality. So, ii) you don't NEED a common denominator anymore. All the parts are there, you could make the phrasing clearer.

 

[Hummingbird25]

i) Your 'therefores' are somewhat backwards. sin(t)/(t+n*pi)>sin(t)/(t+(n+1)*pi) implies the integrand is positive. The integrand being positive DOESN'T imply the inequality. So, ii) you don't NEED a common denominator anymore. All the parts are there, you could make the phrasing clearer.

Okay and thank your for your answer,

Here we go again:

proof:

By applying the limit

sin(t)/(t+n*pi)&gt;sin(t)/(t+(n+1)*pi) the integrand becomes positive for all n. The integral is positive and inequality is thusly true.

How about now?

Best Regards.
Maria.

 

[Dick]

It's not a 'limit', it's an inequality. But good enough.

 

[Hummingbird25]

It's not a 'limit', it's an inequality. But good enough.

thanks

maybe you would look at

https://www.physicsforums.com/showthread.php?t=218339

And maybe comment on my three proofs here ?

Thanks in advance.

Best Regards
Maria.