Answer
Function satisfies conditions for integral test
Series converges
Work Step by Step
$\dfrac {1}{2}+\dfrac {1}{5}+\dfrac {1}{10}+\dfrac {1}{17}+\dfrac {1}{26}\ldots =\sum ^{\infty }_{n=1}\dfrac {1}{n^{2}+1}$
In order integral test can be applied $f\left( x\right) =\dfrac {1}{x^{2}+1}$ must be positive continuous and decreasing for $x>1$
$f'\left( x\right) =\dfrac {-2x}{\left( x^{2}+1\right) ^{2}}$
So integral test can be applied
$$\int ^{\infty }_{1}\dfrac {1}{x^{2}+1}=\arctan x]^{\infty }_{1}=\dfrac {\pi }{4}$$
The integral converges, so the series converges