Question

$\prod _{n=1}^{\infty }\sqrt [n] {\dfrac {3} {4}}=?$

Answer 1

Ilk olarak $$P_k=\prod_{n=1}^k\left(\frac34\right)^{1/n}$$ olarak tanimlayalim. Bizden isten $$\lim\limits_{k\to \infty} P_k$$ degeri. $$P_k=\prod_{n=1}^k\left(\frac34\right)^{1/n}=\left(\frac34\right)^{\sum\limits_{n=1}^k \frac1n}$$ olur ve ayrica $$\lim\limits_{k \to \infty}\sum\limits_{n=1}^k \frac1n \to \infty$$ oldugu bilgisini kullanirsak $(3/4)^x$ fonksiyonu surekli oldugundan $$\lim\limits_{k \to \infty} P_k=\lim\limits_{k \to \infty}\prod_{n=1}^k\left(\frac34\right)^{1/n}=\lim\limits_{k \to \infty}\left(\frac34\right)^{\sum\limits_{n=1}^k \frac1n}=0$$ olur.

Comments

Çok teşekkürler hocam.