# Thread: pre calculus involving limits only

1. ## pre calculus involving limits only

let $\alpha = \displaystyle\lim_{n\to\infty} \left(\frac{1^2 + 2^2 + ..... +n^2}{n^3}\right)$ and
$\beta = \displaystyle\lim_{n\to\infty} \left(\frac{(1^3 - 1^2 )+(2^3 -2^2 )+ ..... +(n^3 - n^2 )}{n^4}\right)$ ,then which of the following is correct
(A) $\alpha = \beta$
(B) $\alpha < \beta$
(C) $4\alpha - 3\beta =0$
(D) $3\alpha - 4\beta =0$

2. Originally Posted by grgrsanjay
let $\alpha = \displaystyle\lim_{n\to\infty} \left(\frac{1^2 + 2^2 + ..... +n^2}{n^3}\right)$ and
$\beta = \displaystyle\lim_{n\to\infty} \left(\frac{(1^3 - 1^2 )+(2^3 -2^2 )+ ..... +(n^3 - n^2 )}{n^4}\right)$ ,then which of the following is correct
(A) $\alpha = \beta$
(B) $\alpha < \beta$
(C) $4\alpha - 3\beta =0$
(D) $3\alpha - 4\beta =0$
$\alpha$ contains a "sum of squares" in the numerator.

$\beta$ contains the difference between a "sum of cubes" and a "sum of squares" in the numerator.

When you express those sums in "closed form", you can evaluate the limits.

You will find the correct answer is one of the 4 listed.

3. Originally Posted by Soroban
Hello, grgrsanjay!

Archie Meade gave you an excellent game plan . . . Could you follow it?

We are expected to know these summation formulas:

. . $1^2 + 2^2 + 3^2 + \hdots + n^2 \:=\:\dfrac{n(n+1)(2n+1)}{6}$

. . $1^3 + 2^3 + 3^3 + \hdots + n^3 \:=\:\dfrac{n^2(n+1)^2}{2}$

$\displaystyle \text{Let }\:\alpha \:=\:\lim_{n\to\infty} \left(\frac{1^2 + 2^2 + ..... +n^2}{n^3}\right) \;\;[1]$

$\displaystyle \text{Let }\:\beta \:=\: \lim_{n\to\infty} \left(\frac{(1^3 - 1^2 )+(2^3 -2^2 )+ \hdots +(n^3 - n^2 )}{n^4}\right) \;\;[2]$

$\text{Then which of the following is correct?}$

. . $(A)\; \alpha \:=\: \beta\qquad (B)\;\alpha \:<\: \beta \qquad (C)\; 4\alpha - 3\beta \:=\:0 \qquad (D)\;3\alpha - 4\beta =0$

In [1], we have: . $\dfrac{1^2+2^2+3^2+\hdots+n^2}{n^3} \;=\;\dfrac{\frac{n(n+1)(2n+1)}{6}}{n^3} \;=\;\dfrac{2n^3+3n^2+n^2}{6n^3}\;=\;\dfrac{1}{3} + \dfrac{1}{2n} + \dfrac{1}{6n^2}$

. . Hence: . $\displaystyle \alpha \;=\;\lim_{n\to\infty}\left(\frac{1}{3} + \frac{1}{2n} + \frac{1}{6n^2}\right) \;=\;\frac{1}{3}$

In [2], we have: . $\dfrac{\frac{n^2(n+1)^2}{4} - \frac{n(n+1)(2n+1)}{6}}{n^4} \;=\;\dfrac{3n^4 + 2n^3 - 3n^2 - 2n}{12n^4} \;=\;\dfrac{1}{4} + \dfrac{1}{6n} - \dfrac{1}{4n^2} - \dfrac{1}{6n^3}$

Hence: . $\displaystyle \beta \;=\;\lim_{n\to\infty}\left(\frac{1}{4} + \frac{1}{6n} - \frac{1}{4n^2} - \frac{1}{6n^3}\right) \;=\;\frac{1}{4}$

The correct answer is $(D).$