Can You Tackle This Advanced Series Summation Challenge?

  • MHB
  • Thread starter Euge
  • Start date
In summary, POTW stands for "Problem of the Week" and is a mathematical or scientific problem presented to the community to solve. The POTW is updated weekly and open to anyone with an interest in mathematics or science. While there are no monetary rewards, solving the POTW can improve problem-solving skills. Solutions are verified through community discussion and peer-review, and the problem poster may also provide their own solution.
  • #1
Euge
Gold Member
MHB
POTW Director
2,073
242
Here is this week's POTW:

-----
Evaluate the sum of the series

$$\sum_{n = 0}^\infty \frac{(-1)^n}{2n+1} \sech\left[\frac{(2n+1)\pi}{2}\right]$$

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
  • #2
No one answered this week's problem. You can read my solution below.

If $S$ is the sum, then $$S = \sum_{n = -\infty}^0 \dfrac{(-1)^n}{-2n+1} \sech\left[\frac{(-2n+1)\pi}{2}\right] = \sum_{n = -\infty}^{-1} \frac{(-1)^n}{2n+1} \sech\left[\frac{(2n+1)\pi}{2}\right]$$ so that $$2S = \sum_{n =-\infty}^\infty \frac{(-1)^n}{2n+1}\sech\left[\frac{(2n+1)\pi}{2}\right] = \frac{1}{2}\sum_{n = -\infty}^\infty \frac{(-1)^n}{n + \frac{1}{2}}\sech\left[\left(n + \frac{1}{2}\right)\pi\right]$$ The function $f(z) = \dfrac{\sech \pi z}{z}$ has $z f(z) \to 0$ as $\lvert z\rvert \to \infty$, so $2S$ is equal to one-half the sum of the residues of $f(z)\pi\sec(\pi z)$ at the singularities of $f$. Now $f$ has simple poles at $z = 0$ and at $z = -i\dfrac{2n+1}{2}$ where $n$ ranges over the integers. The residues of $f(z)\pi \sec(\pi z)$ at $0$ and $-i \dfrac{2n+1}{2}$, respectively, are $\pi$ and $-\dfrac{(-1)^n}{n + \frac{1}{2}}\sech\left[\left(n + \frac{1}{2}\right)\pi\right]$. Therefore, $2S = \dfrac{\pi}{2} - 2S$, or $S = \dfrac{\pi}{8}$.
 

FAQ: Can You Tackle This Advanced Series Summation Challenge?

1. What is POTW?

POTW stands for "Problem of the Week". It is a weekly challenge or puzzle that is designed to test problem-solving skills and critical thinking abilities.

2. Who creates the POTW?

The POTW is typically created by a team of scientists or mathematicians who specialize in creating challenging and thought-provoking problems.

3. How difficult are the POTW challenges?

The difficulty level of the POTW can vary, but they are generally designed to be challenging for individuals with a strong background in science or mathematics. However, they can also be a fun and educational challenge for anyone interested in problem-solving.

4. Are there any rewards for solving the POTW?

Some organizations may offer rewards or recognition for individuals who successfully solve the POTW, but the main purpose is typically for personal growth and development of problem-solving skills.

5. Can I collaborate with others to solve the POTW?

Collaboration is not prohibited, but it is generally encouraged to solve the POTW on your own to truly test your own problem-solving abilities. However, discussing and sharing ideas with others can also be a valuable learning experience.

Back
Top