I'm trying to find closed form for
$$\sum_{k=1}^{n}\sin\frac{1}{k}$$
I typed it in Mathematica 6.0 and WolframAlpha, but no result what i expected.
Any hints will be appreciated, thank you.
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Sign up to join this communityI'm trying to find closed form for
$$\sum_{k=1}^{n}\sin\frac{1}{k}$$
I typed it in Mathematica 6.0 and WolframAlpha, but no result what i expected.
Any hints will be appreciated, thank you.
I doubt you will find a closed form.
Your expression will give a value slightly less than than the harmonic numbers $\sum_{k=1}^{n}\frac{1}{k}$ which do not have a closed form and as $k$ increases the difference will increase towards $0.191899\ldots$, and a value slightly more than $\log n$ and as $k$ increases the difference will fall towards $0.385316\ldots$.
The sum can be expanded in the asymptotic series, several first members being $$ \sum_{k=1}^n\sin\frac{1}{k}= \log n+a+\frac1{2n}-\frac1{12n^3}+O\left(\frac1{n^3}\right), $$ where $$ a=\gamma+\sum_{k=1}^\infty (-1)^k\frac{\zeta(2k+1)}{(2k+1)!} $$ and $\gamma$ is the Euler constant. The value of $a$ is $0.38...$ as written by Henry.