The sum of the series `sum _(n=1) ^(oo) sin ((n!pi)/(720))` is

Options

A. `sin ((pi)/(180)) + sin ((pi)/(360)) + sin ((pi)/(540))`
B. `sin ((pi)/(6)) +sin ((pi)/(30)) + sin ((pi)/(120)) + sin ((pi)/(360))`
C. `sin ((pi)/(6)) + sin ((pi)/(30)) + sin ((pi)/(120)) + sin ((pi)/(360))+sin ((pi)/(720))`
D. `sin ((pi)/(180))+ sin ((pi)/(360))`

Answer

To find the sum of the series \( S = \sum_{n=1}^{\infty} \sin\left(\frac{n! \pi}{720}\right) \), we will analyze the terms of the series step by step.

### Step 1: Write out the first few terms of the series
We start by substituting values of \( n \) into the series.

- For \( n = 1 \):
\[ \sin\left(\frac{1! \pi}{720}\right) = \sin\left(\frac{\pi}{720}\right) \]

- For \( n = 2 \):
\[ \sin\left(\frac{2! \pi}{720}\right) = \sin\left(\frac{2 \pi}{720}\right) = \sin\left(\frac{\pi}{360}\right) \]

- For \( n = 3 \):
\[ \sin\left(\frac{3! \pi}{720}\right) = \sin\left(\frac{6 \pi}{720}\right) = \sin\left(\frac{\pi}{120}\right) \]

- For \( n = 4 \):
\[ \sin\left(\frac{4! \pi}{720}\right) = \sin\left(\frac{24 \pi}{720}\right) = \sin\left(\frac{\pi}{30}\right) \]

- For \( n = 5 \):
\[ \sin\left(\frac{5! \pi}{720}\right) = \sin\left(\frac{120 \pi}{720}\right) = \sin\left{\frac{\pi}{6}\right) \]

- For \( n = 6 \):
\[ \sin\left(\frac{6! \pi}{720}\right) = \sin\left(\frac{720 \pi}{720}\right) = \sin(\pi) = 0 \]

### Step 2: Identify the pattern
From \( n = 6 \) onwards, the factorial grows quickly, and we can see that:

- For \( n \geq 6 \):
\[ n! \text{ is a multiple of } 720, \text{ hence } \sin\left(\frac{n! \pi}{720}\right) = 0 \]

### Step 3: Sum the non-zero terms
Thus, the series reduces to the sum of the first five terms:
\[ S = \sin\left(\frac{\pi}{720}\right) + \sin\left(\frac{\pi}{360}\right) + \sin\left(\frac{\pi}{120}\right) + \sin\left(\frac{\pi}{30}\right) + \sin\left(\frac{\pi}{6}\right) \]

### Step 4: Evaluate each sine term
Now we will evaluate each sine term:

- \( \sin\left(\frac{\pi}{720}\right) \) is a small positive value.
- \( \sin\left(\frac{\pi}{360}\right) \) is also a small positive value.
- \( \sin\left(\frac{\pi}{120}\right) = \sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4} \).
- \( \sin\left(\frac{\pi}{30}\right) = \sin(6^\circ) = \frac{1}{2} \).
- \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

### Step 5: Combine the results
The sum \( S \) can be approximated as:
\[ S \approx \sin\left(\frac{\pi}{720}\right) + \sin\left(\frac{\pi}{360}\right) + \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{1}{2} + \frac{1}{2} \]

### Conclusion
Thus, the sum of the series \( S = \sum_{n=1}^{\infty} \sin\left(\frac{n! \pi}{720}\right) \) converges to a finite value, which is the sum of the first five terms.

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