Is there a typo in the line above? I don't see how this could be true.Focus said:Your series is [tex]x_n=\prod_{i=1}^n \frac{i+9}{2i-1} [/tex]. I'll give you a hint, [tex]\frac{i+9}{2i-1}=\frac{1}{4} \frac{19}{2i-1}[/tex].
Focus said:What happens as i increases (after i=5)?
Good luck
Mark44 said:Is there a typo in the line above? I don't see how this could be true.
I don't see that those two expressions are equal, either. On the right side you get 4i - 17 in the numerator, and 4(2i - 1) in the denominator.Focus said:Yeah sorry that should be[tex] \frac{i+9}{2i-1}=\frac{1}{2}+\frac{1}{4} \frac{19}{2i-1}[/tex]
Mark44 said:I don't see that those two expressions are equal, either. On the right side you get 4i - 17 in the numerator, and 4(2i - 1) in the denominator.
To prove that a function is monotonic, you need to show that its derivative is either always positive or always negative. This means that the function is either always increasing or always decreasing.
Proving monotonicity is important because it helps us understand the behavior of a function. It tells us whether the function is consistently increasing or decreasing, and this information can be useful in making predictions and analyzing data.
A function is bounded if its values do not get infinitely large or infinitely small. In other words, the function is limited and does not have extreme values that approach infinity or negative infinity.
To prove that a function is bounded, you need to show that its values fall within a certain range. This can be done by finding the maximum and minimum values of the function, or by using mathematical techniques such as the squeeze theorem.
Yes, a function can be both monotonic and bounded. For example, a linear function is both monotonic (always increasing or decreasing) and bounded (limited by its y-intercept and slope).