Explain why $$a_{n}$$ monotically decreasing and lower bounded . Determine the limit $$lim_{n} a_{n}$$

$$a_{n}=\frac{5n+1}{7n-2}$$

Question

Explain why $$a_{n}$$ monotically decreasing and lower bounded . Determine the limit $$lim_{n} a_{n}$$

$$a_{n}=\frac{5n+1}{7n-2}$$

terenzreignz · Answer

I suppose we can consider the function 
$$\Large f(x) = \frac{5x+1}{7x-2}$$

terenzreignz · Answer

And find its derivative...
$$\Large f'(x) = \frac{5(7x-2)-7(5x+1)}{(7x-2)^2}=-\frac{17}{(7x-2)^2}$$
Which is negative for all values of x... (except $\large\frac27$)

terenzreignz · Answer

There you have it, a negative derivative...

anonymous · Answer

did we showed in this case that is monotocially decreasing and lower bound ed?

terenzreignz · Answer

Well, its continuous counterpart is (strictly) decreasing for all positive integers...so its discrete case (the sequence) must be monotonically decreasing

anonymous · Answer

ok thx ;)