sequences and series - Let a general term $T_n$ be defined as $T_n =left(frac{1cdot 2cdot 3 cdot 4 cdots n}{1 cdot 3 cdot 5 cdot 7 cdots (2n+1)}right)^2$

Let a general term $T_n$ be defined as

$$T_n =\left(\frac{1\cdot 2\cdot 3 \cdot 4 \cdots n}{1 \cdot 3 \cdot 5 \cdot 7 \cdots (2n+1)}\right)^2$$

Then prove that
$\lim_{n\to\infty}(T_1 + T_2 +\cdots+T_n) \lt \frac{4}{27}.$

I tried finding pattern between terms ..
$T_1=\frac{1}{9} , \frac{T_2}{T_1}=(\frac{2}{5})^2, \frac{T_3}{T_2}=(\frac{3}{7})^2$

but could not think more of how to get a bound on the series.
Any help is appreciated.

Answer

$$\dfrac{T_m}{T_{m-1}}=\left(\dfrac m{2m+1}\right)^2<\left(\dfrac m{2m}\right)^2=\dfrac14$$ for $m>0$

$$\sum_{r=1}^\infty T_r<\sum_{r=1}^\infty T_1\left(\dfrac14\right)^{r-1}=\dfrac{\dfrac19}{1-\dfrac14}=?$$