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 !

Here is a quite elementary way to handle this series:
Set
$$a_n :=\frac 12\cdot \frac 34 \cdots \frac{2n-1}{2n} = \prod_{k=1}^n \frac{2k-1}{2k}$$
Now, set $c_1 = \frac 12$ and for $n \geq 2$
$$c_n := \frac 12\cdot \frac 23 \cdots \frac{2(n-1)}{2n-1}=\frac 12\cdot \prod_{k=2}^{n} \frac{2(k-1)}{2k-1}$$
You can quickly check that
$$c_n < a_n \text{ for } n>1$$
Hence, for $n>1$ you have
$$c_na_n = \frac 14 \prod_{k=2}^{n} \frac{2(k-1)}{2k} \stackrel{telescope}{=}  \frac 14 \frac{2}{2n} = \frac 1{4n} < a_n^2$$
So, for $n>1$ you have
$$a_n > \frac 1{2\sqrt n}$$
Since $\sum_{n=1}^{\infty}\frac 1{2\sqrt n} = \infty$, it follows that $\sum_{n=1}^{\infty}a_n = \infty$, too.
