Test the absolute and conditional convergence
Test the absolute and conditional convergence of the series (for fixed $x$ with $0<x<\pi$): $$ \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^a}. $$
Explanation of Method
Since $0<x<\pi$ ensures $\sin(x/2)\ne0$, the classical closed-form identity for $\sum\sin(n\theta)$ gives a uniform bound $1/\sin(x/2)$ on the partial sums, independent of $N$. This is precisely the hypothesis Dirichlet's test needs, so pairing with the monotone decrease of $n^{-a}$ for $a>0$ secures convergence throughout, with the sharper threshold $a>1$ needed for absolute convergence.
- Derive the closed-form bound on partial sums of $\sin(nx)$.
- Apply Dirichlet's test for $a>0$.
- Determine the absolute-convergence threshold at $a=1$.
Step 1 · Bound the partial sums
$$ \sum_{n=1}^N\sin(nx) = \frac{\sin\left(\frac{Nx}2\right)\sin\left(\frac{(N+1)x}2\right)}{\sin(x/2)}, \qquad \left|\sum_{n=1}^N\sin(nx)\right|\le\frac1{|\sin(x/2)|}. $$
Since $0<x<\pi$ gives $0<x/2<\pi/2$, we have $\sin(x/2)>0$, so this bound is finite and independent of $N$.
Step 2 · Apply Dirichlet's test
For $a>0$, $n^{-a}\downarrow0$, and Step 1 gives bounded partial sums, so by Dirichlet's test the series converges for every $a>0$.
Step 3 · Absolute convergence threshold
Since $x$ is a fixed constant in $(0,\pi)$ not a rational multiple of $\pi$ in general, $|\sin(nx)|$ stays bounded away from $0$ on a positive-density subsequence of $n$, so $\sum|\sin(nx)|/n^a$ compares with $\sum1/n^a$: convergent iff $a>1$. So absolutely convergent for $a>1$, conditionally convergent for $0<a\le1$; for $a\le0$, terms $\sin(nx)/n^a$ don't tend to $0$ (bounded away from zero infinitely often since $x\ne k\pi$), so the series diverges.
Summary Table
| Range of $a$ | Conclusion |
|---|---|
| $a>1$ | Absolutely convergent |
| $0<a\le1$ | Conditionally convergent |
| $a\le0$ | Diverges |
Additional Notes
- The restriction $0<x<\pi$ is essential: at $x=0$ or $x=\pi$, $\sin(nx)\equiv0$ and the series trivially converges to $0$ for every $a$, while the closed-form bound in Step 1 blows up as $x\to0^+$ or $x\to\pi^-$ (since $\sin(x/2)\to0$), reflecting the genuine loss of uniform boundedness at those endpoints.
- This is the sine analogue of Problem 132's cosine series $\sum\cos(n\pi/4)/n^a$, generalizing the fixed angle $\pi/4$ to an arbitrary $x\in(0,\pi)$ — the same Dirichlet's-test mechanism applies verbatim, with the specific angle only affecting the size of the uniform bound, not the threshold at $a=1$.
- This series is a classical building block of Fourier analysis: it is (up to a constant) the imaginary part of $\sum e^{inx}/n^a$, and the case $a=1$ recovers the well-known Fourier series $\sum\sin(nx)/n=\frac{\pi-x}2$ for $0<x<2\pi$, an explicit closed form for the conditionally convergent boundary case.
Result
Absolutely convergent for $a>1$; conditionally convergent for $0<a\le1$: