Prove-x-R-cos-x-cos-2x-cos-nx-n-1-2-n-Z-gt-0-

Question Number 216830 by MrGaster last updated on 22/Feb/25

Prove : \forall x \in R, ∣ \cos x ∣ + ∣ \cos 2 x ∣ + \dots + ∣ \cos n x ∣\geq \frac{n - 1}{2} (n \in Z_{> 0})

Commented by MathematicalUser2357 last updated on 25/Feb/25

Or you could do {n ∣ n \in Z \land n > 0} (or {n ∣ n \in N})

Answered by MrGaster last updated on 23/Feb/25

Prove : \underset{k = 1}{\sum^{n}} ∣ \cos (k π) ∣\geq \frac{n - 1}{2} (n \in Z_{> 0})

Let S_{n} (x) = \underset{k = 1}{\sum^{n}} ∣ \cos (k x) ∣

\Rightarrow S_{n} (x) =∣ \cos (x) ∣ + ∣ \cos (2 x) ∣ + \dots + ∣ \cos (n x) ∣

Consider T_{n} (x) = \underset{k = 1}{\sum^{n}} \cos (k π)

= \frac{\sin (\frac{n x}{2}) \cos (\frac{(n + 1) x}{2})}{\sin (\frac{x}{2})}

\Rightarrow∣ T_{n} (x) ∣\leq \frac{1}{∣ \sin (\frac{x}{2}) ∣}

Let U_{n} (x) = \underset{k = 1}{\sum^{n}} \cos^{2} (k π)

= \frac{n}{2} + \frac{1}{2} \underset{k = 1}{\sum^{n}} \cos (2 k x)

= \frac{n}{2} + \frac{1}{2} (\frac{\sin (n x) \cos ((n + 1) x)}{\sin (x)})

\Rightarrow U_{n} (x) \leq \frac{n}{2} + \frac{1}{2}

∵ ∣ \cos (k x) ∣\geq \cos^{2} (k x)

\Rightarrow S_{n} (x) \geq U_{n} (x)

\Rightarrow S_{n} (x) \geq \frac{n}{2} - \frac{1}{2}

∴ S_{n} (x) \geq \frac{n - 1}{2}

[Q . E . D]

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