Evaluating the infinite series $\sum\limits_{n=1}^\infty(\sin\frac1{2n}-\sin\frac1{2n+1})$

Question

I've been bored and came across in my book a pretty straightforward series problem, namely to determine the convergence of

$$ \sum_{n = 1}^{\infty} \left[\sin\left(1 \over 2n\right) - \sin\left(1 \over 2n + 1\right)\right] $$

Doing so was trivial by rewriting it as an alternating series involving the term $(-1)^k\sin\frac1k$.

Naturally, though, I was curious as to whether this series can be reduced to a simpler closed form in terms of more fundamental constants. Unfortunately I do not immediately know of any techniques of use here or even whether it permits such a 'nice' form. Do any of you?

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I do know from playing with the Euler-Maclaurin sums the value should be something near $0.290674$. As $n\to\infty$ I know the sequence terms behave increasingly like those of the alternating harmonic series (as $\sin x\sim x$ for $|x|\ll1$), which helps explain why it appears relatively near $1-\log2$. I have also found that the difference between it and the alternating harmonic series starting with $1/2$ is near $0.016179$.

I should note that I am a high school student with an amateur interest in recreational math. My knowledge extends only as far as elementary calculus of multiple variables and first-year ordinary and partial differential equations. It may very well be that an obvious approach exists that I've completely missed and so I feel obligated to apologize in advance.

Answer 1

$$\sum_{n=1}^\infty \sin\left({1\over 2n}\right)-\sin\left({1\over 2n+1}\right)=-\sum_{n=1}^\infty\int_{1\over 2n+1}^{1\over 2n}\cos x\, dx$$

These are integrals of a bounded function on the sets

$$\left[{1\over 2n+1},{1\over 2n}\right]$$

The measure of these sets is

$${1\over 2n}-{1\over 2n+1}={1\over 2n(2n+1)}$$

Hence the sum is absolutely convergent by Jensen's inequality since

$$\left|\sum_{n=1}^\infty\int_{1\over 2n}^{1\over 2n+1}\cos x\right|\le\sum_{n=1}^\infty\int_{1\over 2n}^{1\over 2n+1}|\cos x|\le\sum_{n=1}^\infty\int_{1\over 2n+1}^{1\over 2n}1\;dx=\sum_{n=1}^\infty {1\over 2n(2n+1)}.$$

Answer 2

I can't give precise value for the sum of the series, but note that the Mean Value Theorem tells us that it may be written in the form $\sum_{n=1}^{\infty} \frac{ \cos(\theta_{2n})}{2n(2n+1)},$ where each $\theta_{2n} \in (\frac{1}{2n+1},\frac{1}{2n})$ which seems to mean that it differs from $ 1 - \log2 = \sum_{n=1}^{\infty}\frac{1}{2n(2n+1)}$ by something close to $\sum_{n=1}^{\infty}\frac{1}{32n^{4}} = \frac{\pi^{4}}{2880}$, (and is smaller than that sum).