Question

What is the interval of convergence of `sum_1^oo sin(nx)/n`?

Answer 1

`sum_(n=1)^(oo)sin(nx)/n` is conditionally convergent by the alternating series test and the harmonic series.

Explanation:

`sum_(n=1)^oo frac{sin(nx)}{n}`

We can think of this as if it were an alternating series , because the numerator, `sin(nx)`, will only oscillate between `-1` and `1`.

The alternating series test states that we need to prove that:
`color(blue)("I " lim_(n->oo)a_n =0`
`color(blue)(" where "a_n" excludes the alternating part of the series."`

`color(blue)("II "a_n" is monotonically decreasing, " a_(n+1)<=a_n )`

`"Let " a_n=1/n`.

`color(red)("I. ")lim_(n->oo)(1/n)=0`

`color(red)("II. ") 1/(n+1)<=1/n`

Therefore, the sequence is convergent.

To determine whether it has absolute or conditional convergence, test the convergence of:
`sum_(n=1)^(oo)1/n`
We know this is divergent by the harmonic series.

Therefore, `sum_(n=1)^(oo)sin(nx)/n` is conditionally convergent by the alternating series test and the harmonic series.