The period of cos(x + 8x + 27x + ldots + n^3x | vedclass

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Question

(C) We have the expression $\cos(x + 8x + 27x + \ldots + n^3x)$.This can be written as $\cos\left(\sum_{k=1}^{n} k^3 x\right) = \cos\left(x \sum_{k=1}

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$\frac{8 \pi}{n^2(n+1)^2}$

Vedclass · Answer

(C) We have the expression $\cos(x + 8x + 27x + \ldots + n^3x)$.This can be written as $\cos\left(\sum_{k=1}^{n} k^3 x\right) = \cos\left(x \sum_{k=1}^{n} k^3\right)$.Using the formula for the sum of cubes of the first $n$ natural numbers, $\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^2(n+1)^2}{4}$.Thus, the expression becomes $\cos\left(\frac{n^2(n+1)^2}{4} x\right)$.The period of $\cos(kx)$ is $\frac{2\pi}{|k|}$.Here, $k = \frac{n^2(n+1)^2}{4}$.Therefore, the period is $\frac{2\pi}{\frac{n^2(n+1)^2}{4}} = \frac{8\pi}{n^2(n+1)^2}$.

The period of $\cos(x + 8x + 27x + \ldots + n^3x)$ is

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