2

I was trying to find it for some time, but couldn't, so please help me. $ \lim_{x\to\infty} ( \sqrt[100]{(x + 3*1)(x + 3*2)...(x +3*100)} - x)$

kopft
  • 31
  • 4
  • 1
    This limit has been asked many times. See here: https://math.stackexchange.com/questions/1591903/solving-lim-limits-x-to-infty-sqrtnxa-1-xa-2-dots-xa-n-x and here: https://math.stackexchange.com/questions/3491441/limit-lim-limits-x-to-infty-sqrtnxa-1xa-2-xa-n-x and here: https://math.stackexchange.com/questions/922380/evaluate-lim-x-to-infty-leftx-sqrtnx-a-1x-a-2-ldotsx-a-n/ – bjorn93 Jan 06 '20 at 14:55

2 Answers2

2

Set $1/x=h$

$$\lim_{h\to0^+}\dfrac{\sqrt[100]{(1+3h)(1+6h)\cdots(1+300h)}-1}h$$

Now $(1+3h)(1+6h)=1+(3+6)h+O(h^2)$

$(1+3h)(1+6h)(1+9h)=(1+9h+O(h^2))(1+9h)=1+(3+6+9)h+O(h^2)$

Similarly, $(1+3h)(1+6h)\cdots(1+300h)=1+(3+6+9+\cdots+300)h+h^2=1+\dfrac{(3+300)100h}2+O(h^2)$

$\sqrt[100]{(1+3h)(1+6h)\cdots(1+300h)}=\left(1+\dfrac{(3+300)100h}2+O(h^2)\right)^{1/100}=1+\dfrac{303}2h+O(h^2)$

lab bhattacharjee
  • 274,582
2

\begin{gather*} \lim _{x\leadsto \infty }(\sqrt[100]{}( x+3\times 1)( x+3\times 2) ...( x+3\times 100) \ -\ x\\ Taking\ x\ common\ \ from\ all\ brackets\ ,\ \\ \lim _{x\leadsto \infty }( x\sqrt[100]{}( 1+3\times 1/x)( 1+3\times 2/x) ...( 1+3\times 100/x) \ -\ x\\ Apply\ the\ binomial\ series\ expansion\ because\ \frac{1}{x} \leadsto 0\\ \lim _{x\leadsto \infty }( x( 1+\frac{1}{100}\left(\frac{3\times 1}{x} +\frac{3\times 2}{x} +....\frac{3\times 100}{x}\right) \ -\ x\\ \lim _{x\leadsto \infty }\frac{1}{100}\left(\frac{3\times 1}{1} +\frac{3\times 2}{1} +....\frac{3\times 100}{1}\right) \ \ =\ \frac{3}{200}( 100\ \times 101) \ =\frac{303}{2} \ \\ \\ \\ \\ \end{gather*}

Ishraaq Parvez
  • 1,731