A simple problem in a package

Calculus Level 3

Let $x_1, x_2, x_3, \ldots$ be roots of the function $f(x) = \sin{\left( \dfrac{\pi \ln{(x)}}{\ln3} \right)}$ where $1 < x_1 < x_2 < x_3 < \cdots$ .

What is the value of $\sum_{n=1}^{\infty} \dfrac{1}{x_n} \quad ?$

\frac{\pi}{6}

\frac{1}{2}

\ln{\frac{8}{5}}

\frac{\pi^2}{20}

1 solution

First Last
Sep 16, 2016

In order for $\displaystyle\sin(\frac{\pi\ln{x}}{\ln{3}})$ to equal 0, $\displaystyle\frac{\pi\ln{x}}{\ln{3}}$ must equal $k\pi$ , where k is an integer.

$\displaystyle\frac{\pi \ln{x}}{\ln{3}} = k\pi\quad$ so the roots greater than 1 are $\displaystyle x_n = 3^n,\quad n > 0$

In general, $\displaystyle\sum_{n=1}^{\infty}\frac{1}{r^n} = \frac{1}{1-\frac{1}{r}}-1$ so

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{3^n} = \boxed{\frac{1}{2}}$

A simple problem in a package

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