prove that 1/2 * 1/3 * 1/4 * ... ..... ... ..... ..... .... ....... .... ... ....1/99 < 1/2
1/2376 < 1/2
When the numerators are the same, the number with the smaller denominator is larger.
plse explain in detail...
I did
this is no a proper method..
u have done correct , no doubt , but i want step by step
1/2 * x < 1/2 if x < 1
all those fractions out there are less than 1, so the product keeps getting smaller and smaller
To prove that the expression 1/2 * 1/3 * 1/4 * ... * 1/99 < 1/2, we can simplify the expression and compare it to 1/2.
Let's start by simplifying the expression 1/2 * 1/3 * 1/4 * ... * 1/99. We can rewrite it as follows:
1/2 * 1/3 * 1/4 * ... * 1/99 = 1/(2 * 3 * 4 * ... * 99)
Now, let's compare this simplified expression to 1/2.
We can see that the denominators of each fraction in the expression are increasing from 2 to 99. So, the numerator, 1, remains the same.
Next, let's compare the denominator of the simplified expression, which is the product of all the numbers from 2 to 99, i.e., 2 * 3 * 4 * ... * 99, to the denominator 2 of 1/2.
Since all the numbers in the denominator of the simplified expression are greater than 2, we can conclude that the denominator of the expression 1/(2 * 3 * 4 * ... * 99) is greater than 2.
Therefore, 1/(2 * 3 * 4 * ... * 99) < 1/2.
Hence, we have proved that 1/2 * 1/3 * 1/4 * ... * 1/99 < 1/2.