P-i-1-n-a-i-b-i-a-b-b-0-P-

Question Number 5519 by FilupSmith last updated on 17/May/16

P = \underset{i = 1}{\prod^{n}} \frac{a^{i}}{b^{i}} a \neq b, b \neq 0

P = ? ?

Commented by Yozzii last updated on 17/May/16

\frac{a \times a^{2} \times a^{3} \times \dots \times a^{n}}{b \times b^{2} \times b^{3} \times \dots \times b^{n}} = \frac{a^{1 + 2 + 3 + \dots + n}}{b^{1 + 2 + 3 + \dots + n}} = \frac{a^{0.5 n (n + 1)}}{b^{0.5 n (n + 1)}} = {(\frac{a}{b})}^{0.5 n (n + 1)}

P = {(a / b)}^{0.5 n (n + 1)}

Commented by FilupSmith last updated on 17/May/16

P = \frac{a \cdot a^{2} \cdot a^{3} \cdot \dots \cdot a^{n}}{b \cdot b^{2} \cdot b^{3} \cdot \dots \cdot a^{n}}

P = \frac{a^{\frac{1}{2} n (n + 1)}}{b^{\frac{1}{2} n (n + 1)}}

∴ P = {(\frac{a}{b})}^{\frac{n (n + 1)}{2}}

What if n \to \infty ? ?

Commented by Yozzii last updated on 17/May/16

\lim_{n \to \infty} {(a / b)}^{0.5 n (n + 1)} = {\begin{cases} d i v e r g e n t i f a ⩾ - b \\ 0 i f - 1 < (a / b) < 1 \\ d i v e r g e n t i f a > b \end{cases}

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