Question

What is the sum of $n$ terms in the series $\log m + \log \left( \frac{m^2}{n} \right) + \log \left( \frac{m^3}{n^2} \right) + \log \left( \frac{m^4}{n^3} \right) + \dots$?

• A. $\log \left( \frac{n(n-1)}{m(n+1)} \right)$
• B. $\log \left( \frac{m^m}{n^n} \right)$
• C. $\log \left( \frac{m(n-1)}{n(n-1)} \right)$
• D. $\log \left( \frac{m^{n+1}}{n^n} \right)$

Hint:

For series involving logarithms, use the properties of logarithms such as $\log(ab) = \log a + \log b$ to simplify the terms.

Answer 1

The Correct Option is B

We are given the series: \[ \log m + \log \left( \frac{m^2}{n} \right) + \log \left( \frac{m^3}{n^2} \right) + \log \left( \frac{m^4}{n^3} \right) + \dots \] This is a logarithmic series with a pattern where each term involves a power of $m$ in the numerator and a power of $n$ in the denominator. We can use the properties of logarithms to simplify the series: \[ \log m + \log m^2 - \log n + \log m^3 - 2 \log n + \dots \] This simplifies to: \[ \log m^{1+2+3+\dots} - \log n^{0+1+2+\dots} = \log \left( \frac{m^{m}}{n^{n}} \right) \] Thus, the sum of the series is $\log \left( \frac{m^m}{n^n} \right)$.