Limit of an expression

For each of the expression, find \(\sum\limits_{r=1}^{\infty }{{{u}_{r}}}\) if it exists.

1. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{n}{2n+3}\)
2. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{4n}{3n+1}\)
3. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{7{{n}^{2}}+2}{(n+3)(4n+1)}\)
4. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{(6n+5)(5-n)}{(2n+1)(3n+1)}\)
5. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{1+{{e}^{n}}}{2(4-3{{e}^{n}})}\)
6. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{2+{{e}^{-n}}}{(3-7{{e}^{-n}})}\)
7. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{{{n}^{3}}+2}{{{n}^{2}}+3n+1}\)
8. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=\frac{7}{4}-\frac{3}{2(n+1)}-\frac{1}{2(n+2)}\)
9. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=7+\frac{{{3}^{n}}}{{{10}^{n}}}\)
10. \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=-15-\frac{{{2}^{n}}}{{{5}^{n}}}\)