TITLE: How can I get the exact value of this infinite series?
QUESTION [8 upvotes]: I want to compute the exact value of this infinite series 
$$\sum_{n=2}^\infty\arcsin{\left(\dfrac{2}{\sqrt{n(n+1)}(\sqrt{n}+\sqrt{n-1})}\right)}$$
By comparison test, we can get the series is convengence.
I tried to find some hints from  the exact value of $\displaystyle\sum_{n=2}^\infty\arcsin{\left(\dfrac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n^2-1}}\right)}$
,but the split phase method maybe difficult to solve this  question. 
I am not sure whether it has  a closed form.But if not so,how can I evaluate the sum ?

REPLY [2 votes]: Not yet finding useful expressions to which to apply
$$
\arcsin(x)-\arcsin(y)=\arcsin\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right)
$$
for the given sum, I will explain how one can approximate the value of the sum.
First we can compute the series
$$
\begin{align}
&\arcsin\left(\frac2{\sqrt{n(n+1)}(\sqrt{n}+\sqrt{n-1})}\right)\\[6pt]
&\small=n^{-3/2}-\tfrac14n^{-5/2}+\tfrac38n^{-7/2}-\tfrac7{192}n^{-9/2}+\tfrac{17}{128}n^{-11/2}+\tfrac{23}{512}n^{-13/2}+\tfrac{1277}{15360} n^{-15/2}+\tfrac{227}{16384}n^{-17/2}\\
&\small+\tfrac{3669}{32768}n^{-19/2}-\tfrac{22005}{917504}n^{-21/2}+\tfrac{30157}{262144}n^{-23/2}-\tfrac{122493}{10485760}n^{-25/2}+\tfrac{3021997}{37748736}n^{-27/2}+\tfrac{325343}{16777216} n^{-29/2}\\
&\small+\tfrac{1877155}{33554432}n^{-31/2}+\tfrac{839806723}{35433480192}n^{-33/2}+\tfrac{4843811671}{75161927680}n^{-35/2}+\tfrac{62611265}{8589934592}n^{-37/2}+\tfrac{51911966591}{670014898176}n^{-39/2}\\
&\small+\tfrac{585844333}{137438953472}n^{-41/2}+\tfrac{19229654735}{274877906944}n^{-43/2}+\tfrac{1012455521821}{49478023249920}n^{-45/2}+O\left(n^{-47/2}\right)
\end{align}
$$
Then apply the Euler-Maclaurin Sum Formula to get
$$
\begin{align}
&\sum_{k=2}^n\arcsin\left(\frac2{\sqrt{k(k+1)}(\sqrt{k}+\sqrt{k-1})}\right)\\[6pt]
&\small=C-2n^{-1/2}+\tfrac23n^{-3/2}-\tfrac25n^{-5/2}+\tfrac14n^{-7/2}-\tfrac5{36}n^{-9/2}+\tfrac{41}{704}n^{-11/2}-\tfrac{7}{416}n^{-13/2}+\tfrac{29}{1536}n^{-15/2}\\
&\small-\tfrac{209}{4352}n^{-17/2}+\tfrac{148975}{2179072}n^{-19/2}-\tfrac{9931}{172032}n^{-21/2}+\tfrac{758479}{15073280}n^{-23/2}-\tfrac{161979}{1638400}n^{-25/2}+\tfrac{6072637}{56623104}n^{-27/2}\\
&\small+\tfrac{4932997}{30408704}n^{-29/2}-\tfrac{13284961755}{40047214592}n^{-31/2}-\tfrac{10330566383}{9688842240}n^{-33/2}+\tfrac{12196256559}{5368709120}n^{-35/2}+\tfrac{1973513480505}{258234908672}n^{-37/2}\\
&\small-\tfrac{2054154874907}{111669149696}n^{-39/2}-\tfrac{61779979336283}{880468295680}n^{-41/2}+\tfrac{38613309561025709}{206845624975360}n^{-43/2}+O\left(n^{-45/2}\right)
\end{align}
$$
Computing the sum of arcsines up to $n=1000$, then subtracting the non-constant part of the asymptotic expansion, we get the value
$$
C=1.577134230124889513864011795293057977459257178812501378053524
$$