n lim(2^(1)/(2)-2^(1)/(3))(2^(1)/(2)-2^(1)/(5)) ..(2^(1)/(2)-2^(1)/(2n+1)) is equal to :

Question

$\underset{n	o \infty}{lim}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})({2}^{\frac{1}{2}}-{2}^{\frac{1}{5}})\ldots ..({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n+1}})}$ is equal to :

Accepted Answer

Let $L=\underset{n	o \infty}{lim}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})({2}^{\frac{1}{2}}-{2}^{\frac{1}{5}})\ldots ({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n-1}})}$

Here, ${({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})}^{n}<({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})({2}^{\frac{1}{3}}-{2}^{\frac{1}{5}})$
$\ldots ({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n-1}})<{({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n-1}})}^{n}$

$\Rightarrow \underset{n	o \infty}{lim}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})}^{n}<L<\underset{n	o \infty}{lim}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n-1}})}^{n}$

Since, $\underset{n	o \infty}{lim}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})}^{n}=0$ and $\underset{n	o \infty}{lim}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n-1}})}^{n}=0$

$	herefore L=0$

Solutions · 1