What Real Number and Positive Limit Solve This Integral Ratio Challenge?

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    2016
In summary, the POTW (Problem of the Week) is a weekly challenge or question given to individuals in the science field to solve or answer. It is chosen by a team of scientists or educators and there is usually a specific solution, although different approaches may be accepted. Participation is open to anyone interested in science, and while there may be rewards or recognition, the main reward is the satisfaction of solving a challenging problem and improving critical thinking skills.
  • #1
Ackbach
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Here is this week's POTW:

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Find a real number $c$ and a positive number $L$ for which
$$\displaystyle\lim_{r\to\infty} \frac{\displaystyle r^c \int_0^{\pi/2} x^r \sin(x) \,dx}{\displaystyle \int_0^{\pi/2} x^r \cos(x) \,dx} = L.$$

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  • #2
Congratulations to Kiwi for his correct solution, which follows:

I choose to let r tend to infinity through the sequence of integer values r = {3,7,11, ..., 3+4t ...}.

Let
\(I= \int x^r \sin(x) \,dx\)

and
\(J = \int x^r \cos(x) \,dx\)

so
\(\displaystyle\lim_{r\to\infty} \frac{ r^c \int_0^{\pi/2} x^r \sin(x) \,dx}{ \int_0^{\pi/2} x^r \cos(x) \,dx} = \displaystyle\lim_{r\to\infty} \frac{r^c ^{\pi/2}_0}{[J]^{\pi/2}_0}L.\)

Now by repeated application of integration by parts (remembering r = 3 mod 4):
\(J=x^r \sin(x) + r x^{r-1} \cos(x) - r^2 x^{r-2} \sin(x) - r^3 x^{r-3} \cos(x) +
... + r^{r-3} x^3 \sin(x) + r^{r-2} x^2 \cos(x) - r^{r-1} x \sin(x) - r^r \cos(x)\)

\([J]^{\pi/2}_{0}=
(x^r \sin(x) - r^2 x^{r-2} \sin(x) + ... + r^{r-3} x^3 \sin(x) - r^{r-1} x \sin(x))|_{x=\pi/2} - \displaystyle\lim_{x\to 0} \frac rx
(x^r \cos(x) - r^2 x^{r-2} \cos(x) + ... + r^{r-3} x^3 \cos(x) - r^{r-1} x \cos(x))\)

\(\therefore [J]^{\pi/2}_{0}=
(x^r - r^2 x^{r-2} + ... + r^{r-3} x^3 - r^{r-1} x )|_{x=\pi/2} - \displaystyle\lim_{x\to 0} \frac rx
(x^r - r^2 x^{r-2} + ... + r^{r-3} x^3 - r^{r-1} x )\)

\(\therefore [J]^{\pi/2}_{0}=
(x^r - r^2 x^{r-2} + ... + r^{r-3} x^3 - r^{r-1} x )|_{x=\pi/2} - 1\) \(\tag{1}\)

Similarly
\(I=-x^r \cos(x) + r x^{r-1} \sin(x) + r^2 x^{r-2} \cos(x) - r^3 x^{r-3} \sin(x) + ...
-r^{r-3}x^3 \cos(x) + r^{r-2} x^2 \sin(x) + r^{r-1} x \cos(x) - r^r \sin(x)\)

noting that the cos terms are all zero for x=0 or x=pi/2 and that the sin terms are not singular and are equal to zero as x approaches 0.
\(^{\pi/2}_{0}= \frac rx
(x^r \sin(x) - r^2 x^{r-2} \sin(x) + ... + r^{r-3} x^3 \sin(x) - r^{r-1} x \sin(x))|_{x=\pi/2}\)

\(\therefore ^{\pi/2}_{0}= \frac rx
(x^r - r^2 x^{r-2} + ... + r^{r-3} x^3 - r^{r-1} x )|_{x=\pi/2}\) \(\tag{2}\)

So finally, using (1) and (2):
\(\displaystyle\lim_{r \to \infty} \frac{r^c^{\pi/2}_0}
{ [J]^{\pi/2}_0}=\displaystyle\lim_{r \to \infty}\frac {r^c(\frac rx(x^r - r^2 x^{r-2} + ... + r^{r-3} x^3 - r^{r-1} x ))|_{x=\pi/2}}{(x^r - r^2 x^{r-2} + ... + r^{r-3} x^3 - r^{r-1} x )|_{x=\pi/2}-1}=\displaystyle\lim_{r \to \infty}\frac{r^cr2}{\pi}\)

So
\(\displaystyle\lim_{r\to\infty} \frac{ r^c \int_0^{\pi/2} x^r \sin(x) \,dx}{ \int_0^{\pi/2} x^r \cos(x) \,dx} =\frac 2\pi\) if c = -1 and cannot be a positive real number for any other value of c.
 

FAQ: What Real Number and Positive Limit Solve This Integral Ratio Challenge?

1. What is the POTW?

The POTW stands for "Problem of the Week." It is a weekly challenge or question given to students or professionals in the science field to solve or answer. It allows individuals to apply their knowledge and critical thinking skills to real-world problems.

2. How is the POTW chosen?

The POTW is usually chosen by a team of scientists or educators who come up with a question or problem that is relevant and challenging. They may also take suggestions from the community or select a topic that is currently being researched or discussed in the scientific community.

3. Is there a specific solution to the POTW?

Yes, there is usually a specific solution to the POTW. However, there may be multiple ways to arrive at the solution, and different approaches may be accepted as long as they are scientifically sound and logical.

4. Can anyone participate in the POTW?

Yes, anyone with a background or interest in science can participate in the POTW. It is open to students, professionals, and even those who are passionate about science as a hobby. It is a great way to challenge oneself and learn new concepts.

5. Are there any rewards for solving the POTW?

Some organizations or schools may offer rewards or recognition for individuals who successfully solve the POTW. However, the main reward is the satisfaction of solving a challenging problem and improving one's critical thinking and problem-solving skills.

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