Using Dirichlet's test, prove the convergence of: sum
Using Dirichlet's test, prove the convergence of: $$ \sum_{n=1}^{\infty} a_n \cos n \alpha, $$ where $a_{n-1}>a_n>0$ and $\lim_{n\to\infty}a_n=0$.
Explanation of Method
Here the roles are reversed from the typical setup: $\{a_n\}$ itself is the given decreasing-to-zero sequence, and $b_n=\cos n\alpha$ needs bounded partial sums. The same closed-form identity used for sines (Problem 114) has a cosine analogue, giving a uniform bound on $\sum\cos n\alpha$ whenever $\alpha\ne2k\pi$ — precisely the case excluded is where $\cos n\alpha\equiv1$ and the partial sums genuinely diverge.
- Handle the exceptional case $\alpha=2k\pi$ separately.
- Bound the partial sums $B_N=\sum\cos n\alpha$ for $\alpha\ne2k\pi$ using the closed-form identity.
- Apply Dirichlet's test using the given monotonicity and limit of $\{a_n\}$.
Step 1 · Exceptional case $\alpha=2k\pi$
If $\alpha=2k\pi$, $\cos n\alpha=1$ for every $n$, so the series becomes $\sum a_n$, an ordinary positive decreasing series whose convergence is not guaranteed by the given hypotheses alone (it depends on the specific rate of decay of $a_n$).
Step 2 · Bound the partial sums for $\alpha\ne2k\pi$
Using the identity $\displaystyle\sum_{n=1}^{N}\cos n\theta = \frac{\sin\!\left(\frac{(N+1)\theta}2\right)\cos\!\left(\frac{N\theta}2\right)}{\sin(\theta/2)}-\frac12$, valid since $\sin(\alpha/2)\ne0$:
$$ |B_N| \le \frac1{|\sin(\alpha/2)|} + \frac12, $$
a bound independent of $N$.
Step 3 · Apply Dirichlet's test
Given: $a_n$ is monotonically decreasing ($a_{n-1}>a_n$) and $a_n\to0$. Combined with the bounded partial sums $B_N$ of $\cos n\alpha$ from Step 2, all conditions of Dirichlet's test are satisfied for $\alpha\ne2k\pi$, so $\sum a_n\cos n\alpha$ converges.
Summary Table
| Condition | Conclusion |
|---|---|
| $\alpha\ne2k\pi$ | Converges by Dirichlet's test |
| $\alpha=2k\pi$ | Reduces to $\sum a_n$; not automatically resolved |
Additional Notes
- This problem is the "template" theorem underlying several concrete instances elsewhere in this collection: Problem 116's $\sum\frac{\ln^2n}n\sin\frac{n\pi}4$ and Problem 117's cosine series are both special cases of this general result, once their coefficient sequences are shown to be eventually monotonic decreasing to zero.
- The exceptional case $\alpha=2k\pi$ is genuinely different in kind, not just degree: there, all the oscillation disappears (every term has the same sign), so the problem reduces entirely to whether the plain positive series $\sum a_n$ converges — no longer covered by Dirichlet's test at all.
- Since Dirichlet's test only asks for boundedness of $\sum\cos n\alpha$'s partial sums (not any specific numeric bound), the exact value $\frac1{|\sin(\alpha/2)|}+\frac12$ is somewhat incidental — what matters structurally is simply that $\alpha$ avoids the degenerate multiples of $2\pi$.
Result
For $\alpha\ne2k\pi$, the series converges by Dirichlet's test: