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Let $S_n=\sum_{k=1}^{n}\frac{\sin\frac{k\pi}{25}}{k}$,how many positive $S_n$ are in $S_1,S_2,...S_{100}$

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I think it is positive for all values of n.As Prasad G mentioned $sin(\frac{k\pi}{25})$ is positive for the values between 25-50 and 50-75. However, we will get positive sum when n is 55 for example, because of symmetry of sin function(consider the sin on the unit circle and observe that it is symmetric wrt x-axis) and the term k on the denominator. Negative terms will be divided by bigger k so sum turns out to be positive eventually.

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Let $S_n=\sum_{k=1}^{n}\frac{\sin\frac{k\pi}{25}}{k}$,how many positive $S_n$ are in $S_1,S_2,...S_{100}$

$sin A$ is positive when $A$ is lies between $0$ and $\pi$.

There fore, $\sin\frac{k\pi}{25}$ is positive when $0 \leq \frac{k\pi}{25} \leq \pi$ and $2\pi < \frac{k\pi}{25} \leq 3\pi$.

this implies that, $\sin\frac{k\pi}{25}$ is positive when $0 \leq k \leq 25$ and $50 < k \leq 75$

So, you can find your problem from this onwards....

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    thank you.yeah all of these are positive.Since $sin\frac{(k+25)\pi}{25} = -sin\frac{k\pi}{25}$ and \frac{1}{k+25} < \frac{1}{k}.2012-07-13