# Thread: limits of sequences: u_n = [(2n^3 - 4n^2 + 5)/(10n^3 + 100)]*2^{-n), etc

1. ## limits of sequences: u_n = [(2n^3 - 4n^2 + 5)/(10n^3 + 100)]*2^{-n), etc

hello

I really troubled with the following questions:

$\mbox{1. }\, u_n\, =\, \left(\dfrac{2n^3\, -\, 4n^2\, +\, 5}{10n^3\, +\, 100}\right)\, \cdot\, 2^{-n}$

$\mbox{2. }\, w_n\, =\, \left(\dfrac{3n\, +\, 2}{4n^2\, +\, 8n\, +\, 5}\right)\, \cdot\, \left(\dfrac{(1\, -\, n)^3}{(14\, -\, 5n)^2}\right)$

$\mbox{3. }\, e_n\, =\, \dfrac{2^n\, +\, 3^{n-1}\, +\, 5^{2n+2}}{4^{n-7}\, +\, 5^{2n}}$

$\mbox{4. }\, b_n\, =\, \dfrac{\left(\frac{2}{3}\right)^n}{\left(\frac{1}{ 2}\right)^n\, +\, \left(\frac{9}{10}\right)^n}$

$\mbox{5. }\, S_n\, =\, 4\, \sqrt{\strut n\, +\, 3\,}\, -\, \sqrt{\strut n\, -\, 1\,}\, -\, 5\, \sqrt{\strut n\, +\, 7\,}\, +\, 2\, \sqrt{\strut n\, -\, 3\,}$

I need to find the limit of each sequence.

My main concrn is thos 3 last sequence.

2. Originally Posted by someone111888
hello

I really troubled with the following questions:

$\mbox{1. }\, u_n\, =\, \left(\dfrac{2n^3\, -\, 4n^2\, +\, 5}{10n^3\, +\, 100}\right)\, \cdot\, 2^{-n}$

$\mbox{2. }\, w_n\, =\, \left(\dfrac{3n\, +\, 2}{4n^2\, +\, 8n\, +\, 5}\right)\, \cdot\, \left(\dfrac{(1\, -\, n)^3}{(14\, -\, 5n)^2}\right)$

$\mbox{3. }\, e_n\, =\, \dfrac{2^n\, +\, 3^{n-1}\, +\, 5^{2n+2}}{4^{n-7}\, +\, 5^{2n}}$

$\mbox{4. }\, b_n\, =\, \dfrac{\left(\frac{2}{3}\right)^n}{\left(\frac{1}{ 2}\right)^n\, +\, \left(\frac{9}{10}\right)^n}$

$\mbox{5. }\, S_n\, =\, 4\, \sqrt{\strut n\, +\, 3\,}\, -\, \sqrt{\strut n\, -\, 1\,}\, -\, 5\, \sqrt{\strut n\, +\, 7\,}\, +\, 2\, \sqrt{\strut n\, -\, 3\,}$

I need to find the limit of each sequence.

My main concrn is thos 3 last sequence.
Unfortunately, you have not shown us any work of your own. You must have something or it is entirely unfair to have given you these problems on an assignment.

For example:
1) Who cares what n^2 does if there is an n^3 in there?
2) Does it really matter what (1/8)^n does if there us a (8/9)^n right next to it?
3) If part of an expression converges to a finite value other than zero, and its neighbor converges to zero, what can we say of the entire expression?

No hard and fast rules, here. Waiting for you to show us where you are.

Try to remember that Convergence has nothing to do with a couple of terms (read: Finite Number of Terms) at the beginning. We are examining tail behavior.

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