Fourier series--showing converges to pi/16

[Dustinsfl]

When a Fourier series contains only sine and cosine terms, evaluating the series isn't too difficult.

However, I want to show a Fourier series with sine and sinh converges to \(\frac{\pi}{16}\).
\[
T(50, 50) = \sum_{n = 1}^{\infty}
\frac{\sin\left(\frac{\pi(2n - 1)}{2}\right)
\sinh\left(\frac{\pi(2n - 1)}{2}\right)}
{(2n - 1)\sinh[(2n - 1)\pi]} = \frac{\pi}{16}
\]
Since this series contains sinh, I am not sure how to evaluate it.

 

[Deveno]

My thought: use

$\text{sinh}(x) = -i \sin(ix)$

 

[Dustinsfl]

My thought: use

$\text{sinh}(x) = -i \sin(ix)$

I think I will still have some trouble though. I end up with:
\[
\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}}{2n - 1}\frac{\sin\left[\frac{i\pi}{2}(2n-1)\right]}{\sin\left[i\pi(2n-1)\right]}
\]

 

[Opalg]

When a Fourier series contains only sine and cosine terms, evaluating the series isn't too difficult.

However, I want to show a Fourier series with sine and sinh converges to \(\frac{\pi}{16}\).
\[
T(50, 50) = \sum_{n = 1}^{\infty}
\frac{\sin\left(\frac{\pi(2n - 1)}{2}\right)
\sinh\left(\frac{\pi(2n - 1)}{2}\right)}
{(2n - 1)\sinh[(2n - 1)\pi]} = \frac{\pi}{16}
\]
Since this series contains sinh, I am not sure how to evaluate it.

The sin function isn't really there at all, because $\sin\left(\frac{\pi(2n - 1)}{2}\right) = \sin\left(\bigl(n-\frac12\bigr)\pi\right) = (-1)^{n-1}$. Also, you can use the identity $\sinh(2x) = 2\sinh x\cosh x$, to rewrite the sum as \(\displaystyle \sum_{n = 1}^{\infty}\frac{(-1)^n}{2(2n-1)\cosh\left(\bigl(n-\frac12\bigr)\pi\right)}.\) That looks a bit less complicated than the original series, but I still don't see how to sum it. It clearly converges very fast, because the cosh term in the denominator will get very large after the first few terms. A numerical check shows that the sum of the first three terms is $0.1968518...$, which is very close to $\pi/16$. But that isn't a proof!

 

[blue_raver22]

I would first clarify what is meant by "converges to \(\frac{\pi}{16}\)". Is this referring to the limit of the series as the number of terms approaches infinity, or to the value of the series at a specific point? Additionally, I would want to know the context and purpose of this series, as it may impact the approach to evaluating it.

That being said, evaluating a Fourier series with sinh terms may be more challenging than one with just sine and cosine terms. One approach could be to use the properties of sinh to rewrite the series in terms of exponential functions, which may make it easier to evaluate. Alternatively, numerical methods such as approximation or integration techniques could be used to estimate the value of the series.

In general, the convergence of a series depends on the behavior of its terms, so understanding the properties of the terms in this series will be crucial in determining its convergence. Overall, evaluating this Fourier series may require a combination of mathematical techniques and careful analysis to determine its convergence to \(\frac{\pi}{16}\).