Using the Weierstrass test, show that the series sum
Using the Weierstrass test, show that the series $\displaystyle\sum_{n=1}^{\infty}\frac{\cos nx}{n^a}$ and $\displaystyle\sum_{n=1}^{\infty}\frac{\sin nx}{n^a}$ converge uniformly on $(-\infty,\infty)$ for $a>1$.
Explanation of Method
Both $|\cos nx|$ and $|\sin nx|$ are bounded by $1$ for every $x$ and $n$, so each series is dominated termwise by the convergent $p$-series $\sum1/n^a$ ($a>1$) — the textbook setup for the Weierstrass M-test, with a bound that requires no dependence on $x$ whatsoever.
- Bound each term by $1/n^a$, uniformly in $x$.
- Confirm $\sum1/n^a$ converges for $a>1$.
- Apply the Weierstrass M-test to conclude uniform (and absolute) convergence.
Step 1 · Uniform bound
For every $x\in\mathbb R$ and $n\ge1$: $|\cos nx|\le1$ and $|\sin nx|\le1$, so $\left|\dfrac{\cos nx}{n^a}\right|\le\dfrac1{n^a}$ and $\left|\dfrac{\sin nx}{n^a}\right|\le\dfrac1{n^a}$.
Step 2 · Majorant series
$\sum_{n=1}^{\infty}M_n=\sum_{n=1}^{\infty}1/n^a$ converges, being a $p$-series with $p=a>1$.
Step 3 · Weierstrass M-test
Both conditions of the M-test are satisfied with $M_n=1/n^a$, so both series converge uniformly and absolutely on all of $\mathbb R$.
Summary Table
| Series | Majorant $M_n$ |
|---|---|
| $\sum\cos(nx)/n^a$ | $1/n^a$ |
| $\sum\sin(nx)/n^a$ | $1/n^a$ |
Additional Notes
- The threshold $a>1$ here matches exactly the absolute-convergence threshold found pointwise in Problems 132 and 133, but the M-test additionally certifies uniform convergence on all of $\mathbb R$ — a strictly stronger statement not addressed by the earlier pointwise analysis.
- For $a=1$, the series still converge pointwise (for $x$ away from multiples of $2\pi$, via Dirichlet's test, as used in Problems 132–133), but the Weierstrass M-test cannot apply since its natural majorant $\sum1/n$ diverges — uniform convergence genuinely fails near the excluded points in that boundary case.
- These trigonometric series with $a>1$ are, up to notation, the real and imaginary parts of the complex power series $\sum e^{inx}/n^a$ evaluated on the unit circle — the polylogarithm function $\mathrm{Li}_a(e^{ix})$, a classical special function in analytic number theory.
Result
Both series converge uniformly and absolutely on $\mathbb R$ for $a>1$: