Find the sum of the convergent series. sum_{n=1}^inftyfrac{1}{9n^2+3n-2}

Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
$f(x)={\tan }^{-1}4x,a=0$

Consider the following infinite series.
a.Find the first four partial sums $S_1,S_2,S_3,$ and $S_4$ of the series.
b.Find a formula for the nth partial sum $S_n$ of the indinite series.Use this formula to find the next four partial sums $S_5,S_6,S_7$ and $S_8$ of the infinite series.
c.Make a conjecture for the value of the series.
$\sum _{k=1}^{\mathrm{\infty }}\frac{2}{(2k-1)(2k+1)}$

Find whether the series diverges and its sum:
${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{(-1)}^{n+1}\frac{3}{5^n}}$

Compute each of these double sums:
a) ${\displaystyle \sum _{i=1}^2\sum _{i+1}^3(i+j)}$
b) ${\displaystyle =\sum _{i=1}^2(i+1+i+2+j+3)}$

Evaluate
${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2{2}^n}}$

Does the sum ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\sqrt{n!}}{(3+\sqrt{1})(3+\sqrt{2})\dots (3+\sqrt{n})}}$ converge or diverge?

What options do I have for this series? No idea how to do it.
${\displaystyle \sum _{k=1}^{\mathrm{\infty }}\frac{{\cos }^2\left(k\right)}{4k^2-1}}$