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Examples 4.2.4(a):

The series converges.

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Because | sin(x) | 1 for all real number x, we have that

1 / 2 n sin(n) | | 1 / 2 n |
But is a special case of the geometric series, which converges. Hence, by the comparison test, the original series converges absolutely.

Note that, formally, we could say: Since

| 1 / 2 n sin(n) | | 1 / 2 n |
for all n. We also have
| | <
"Therefore", the series converges absolutely.
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