The period of the function f(x) = e^{x - [x] | vedclass

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Question

(A) The function is given by $f(x) = e^{\{x\} + |\cos \pi x| + |\cos 2\pi x| + \dots + |\cos n\pi x|}$,where $\{x\} = x - [x]$ is the fractional part

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$1$

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(A) The function is given by $f(x) = e^{\{x\} + |\cos \pi x| + |\cos 2\pi x| + \dots + |\cos n\pi x|}$,where $\{x\} = x - [x]$ is the fractional part function.$1$. The period of the fractional part function $\{x\}$ is $T_0 = 1$.$2$. For each term of the form $|\cos(k\pi x)|$,the period $T_k$ is given by $\frac{\pi}{k\pi} = \frac{1}{k}$ for $k = 1, 2, \dots, n$.$3$. The period of the function $f(x)$ is the least common multiple $(LCM)$ of the periods of its individual components: $T = 	ext{LCM}(T_0, T_1, T_2, \dots, T_n) = 	ext{LCM}(1, 1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n})$.$4$. The $LCM$ of a set of fractions $\frac{a}{b}, \frac{c}{d}, \dots$ is given by $\frac{	ext{LCM}(a, c, \dots)}{	ext{GCD}(b, d, \dots)}$.$5$. Thus,$T = \frac{	ext{LCM}(1, 1, 1, \dots, 1)}{	ext{GCD}(1, 2, 3, \dots, n)} = \frac{1}{1} = 1$.Therefore,the period of the function is $1$.

The period of the function $f(x) = e^{x - [x] + |\cos \pi x| + |\cos 2\pi x| + \dots + |\cos n\pi x|}$ (where $[.]$ denotes the greatest integer function) is:

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