In
$$\frac{Gt}{j} = \frac{Gj}{R}\Big ( \frac{1}{6}+\frac{2}{\pi^2}
\sum_{n=1}^{\infty}\frac{(-1)^2}{n^2}
\exp(\frac{-n^2\pi^2 Rt}{j^2})
\Big )$$
How to find the range of $Rt/j^2$ for which the equation is valid?
In
$$\frac{Gt}{j} = \frac{Gj}{R}\Big ( \frac{1}{6}+\frac{2}{\pi^2}
\sum_{n=1}^{\infty}\frac{(-1)^2}{n^2}
\exp(\frac{-n^2\pi^2 Rt}{j^2})
\Big )$$
How to find the range of $Rt/j^2$ for which the equation is valid?
does $j = \sqrt{-1}$ ?
I'm assuming it does. Thus we have
$b_n = \dfrac{e^{n^2\pi^2 R t}}{n^2}$
is the underlying sequence in the alternating series.
In order for the series to converge this sequence must have two properties
i) $b_n$ is ultimately a decreasing sequence
ii) $\lim \limits_{n \to \infty}~b_n = 0$
we need the sign of $R t$
if $R t > 0$ then this sequence tends to infinity
if $R t \leq 0$ then it tends to 0.
additionally if $R t \leq 0$ then the sequence is decreasing.
So it looks like your answer is the formula is valid for $R t \leq 0$
no, it is a positive veriable
yes,
The entire solution to a partial differential equation is
$$y(x,t) = \frac{Gt}{j}+\frac{Gj}{D}\Bigg(\frac{3x^2-j^2}{6j^2}-\frac{2}{\pi^2}\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}\exp\Big(\frac{-n^2\pi^2 Rt}{j^2}\Big) \cos\frac{n\pi x}{j}\Bigg)$$
The above equation is for the case at $x=0$. At a constant $R/j^2$, $y$ increases with $t$. The purpose is to find up to what value of $Rt/j^2$, $y$ remains zero.