Question

Find the limit of the (An)n>=1
A(n)=f(1)*f(2)*...*f(n)
f:R\{-3/2}->R;f(x)=(x+1)/(2x+3)

Answer 1

I've used the Stoltz-Cesaro theorem and gave me the value 1/2
But my answer is wrong

Answer 2

@mathmale

Answer 3

Since
\[

\lim_{n->\infty} f(n)=\frac 1 2

\]
There is exist N so n>N
\[
0< f(n) < \frac 3 4\\
0\le A(n)\le f(1)*f(2)* \cdots f(N)* f(N+1) *fN(+2)*\cdots f(n)\le \\
f(1)*f(2)* \cdots f(N)\left ( \frac 34 \right)^{n-N}=C_N \left ( \frac 34 \right)^{n-N}
\]
If n goes to infinity, then A(n) goes to zero, since
\[
\lim_{n->\infty} C_N \left ( \frac 34 \right)^{n-N}=0
\]