<s>[INST] Sum of series with changing ratio I have a series of the type
$$\frac{1}{2} + \frac{1}{2} \cdot \frac{3}{4} + \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} + \dots $$
The series has a changing ratio $= \frac{2k-1}{2k}$.
The ratio in itself is increasing but always less than $1$.
Q. Is the sum of the series finite? What is the sum of the series?
I have programmed it in Matlab, but the result has not converged even after $10^{13}$ terms.
The difference keeps on decreasing, but the rate of decrease in difference is also decreasing.
Any reference to literature is also appreciated.
Thank you for your time !
 [/INST] By Stirling's approximation, $n!\approx\sqrt{2\pi n}\left(\frac ne\right)^n$. On the other hand,\begin{align}\frac12\times\frac34\times\cdots\times\frac{2n-1}{2n}&=\frac{(2n-1)!!}{(2n)!!}\\&=\frac{(2n-1)!!\times2\times4\times\cdots\times(2n)}{2\times4\times\cdots\times(2n)\times2\times4\times\cdots\times(2n)}\\&=\frac{(2n)!}{2^{2n}(n!)^2}.\end{align}Putting all this together, you get that\begin{align}\frac12\times\frac34\times\cdots\times\frac{2n-1}{2n}&\approx\frac{\sqrt{4\pi n}\left(\frac{2n}e\right)^{2n}}{2^{2n}2\pi n\left(\frac ne\right)^{2n}}\\&=\frac1{\sqrt{\pi n}}.\end{align}Since the series $\displaystyle\sum_{n=1}^\infty\frac1{\sqrt{\pi n}}$ diverges, then so does your series.
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