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

1 Answer
Anjali G
Apr 6, 2017

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.

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