To solve the given series, we need to compute the sum of each term in the sequence:
[tex]\[
\frac{1}{3 \cdot 5 \cdot 7} + \frac{1}{5 \cdot 7 \cdot 9} + \frac{1}{7 \cdot 9 \cdot 11} + \frac{1}{9 \cdot 11 \cdot 13} + \ldots + \frac{1}{23 \cdot 25 \cdot 27}
\][/tex]
Let's calculate each term individually:
1. Calculate the first term:
[tex]\[
\frac{1}{3 \cdot 5 \cdot 7} = \frac{1}{105} \approx 0.009523809523809525
\][/tex]
2. Calculate the second term:
[tex]\[
\frac{1}{5 \cdot 7 \cdot 9} = \frac{1}{315} \approx 0.0031746031746031746
\][/tex]
3. Calculate the third term:
[tex]\[
\frac{1}{7 \cdot 9 \cdot 11} = \frac{1}{693} \approx 0.001443001443001443
\][/tex]
4. Calculate the fourth term:
[tex]\[
\frac{1}{9 \cdot 11 \cdot 13} = \frac{1}{1287} \approx 0.000777000777000777
\][/tex]
5. Calculate the fifth term:
[tex]\[
\frac{1}{11 \cdot 13 \cdot 15} = \frac{1}{2145} \approx 0.0004662004662004662
\][/tex]
6. Calculate the sixth term:
[tex]\[
\frac{1}{13 \cdot 15 \cdot 17} = \frac{1}{3315} \approx 0.00030165912518853697
\][/tex]
7. Calculate the seventh term:
[tex]\[
\frac{1}{15 \cdot 17 \cdot 19} = \frac{1}{4845} \approx 0.0002063983488132095
\][/tex]
8. Calculate the eighth term:
[tex]\[
\frac{1}{17 \cdot 19 \cdot 21} = \frac{1}{6807} \approx 0.00014742739200943535
\][/tex]
9. Calculate the ninth term:
[tex]\[
\frac{1}{19 \cdot 21 \cdot 23} = \frac{1}{9237} \approx 0.00010896807235480004
\][/tex]
10. Calculate the tenth term:
[tex]\[
\frac{1}{21 \cdot 23 \cdot 25} = \frac{1}{12075} \approx 0.00008281573498964803
\][/tex]
11. Calculate the eleventh term:
[tex]\[
\frac{1}{23 \cdot 25 \cdot 27} = \frac{1}{15525} \approx 0.0000644122383252818
\][/tex]
Now, sum all the computed terms:
[tex]\[
0.009523809523809525 + 0.0031746031746031746 + 0.001443001443001443 + 0.000777000777000777 + 0.0004662004662004662 + 0.00030165912518853697 + 0.0002063983488132095 + 0.00014742739200943535 + 0.00010896807235480004 + 0.00008281573498964803 + 0.0000644122383252818
\][/tex]
Adding these values together, we get:
[tex]\[
0.009523809523809525 + 0.0031746031746031746 + 0.001443001443001443 + 0.000777000777000777 + 0.0004662004662004662 + 0.00030165912518853697 + 0.0002063983488132095 + 0.00014742739200943535 + 0.00010896807235480004 + 0.00008281573498964803 + 0.0000644122383252818 \approx 0.0162962962962963
\][/tex]
Thus, the sum of the series is approximately:
[tex]\[
0.0162962962962963
\][/tex]
