Determine whether the geometric series is convergent or divergent. If it is conv

Which of the answers is an arithmetic sequence?
a) 2, 4, 8, 16, 32
b) 3, 6, 9, 15, 24
c) 2, 5, 7, 12, 19
d). 6, 13, 20, 27, 34

Find the radius of convergence, R, of the series.
$\sum _{n=1}^{\mathrm{\infty }}\frac{(2x+9{)}^n}{n^2}$
Find the interval, I, of convergence of the series.

a) Find the Maclaurin series for the function
$f(x)=\frac{1}{1}+x$
b) Use differentiation of power series and the result of part a) to find the Maclaurin series for the function
$g(x)=\frac{1}{(x+1{)}^2}$
c) Use differentiation of power series and the result of part b) to find the Maclaurin series for the function
$h(x)=\frac{1}{(x+1{)}^3}$
d) Find the sum of the series
$\sum _{n=3}^{\mathrm{\infty }}\frac{n(n-1)}{2n}$
This is a Taylor series problem, I understand parts a - c but I do not understand how to do part d where the answer is $\frac{7}{2}$

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 3-4+16/3-64/9+........

Let x be any positive real number, and define a sequence ${\displaystyle \left[a_n\right]}$ by
${\displaystyle a_n=\frac{\left[x\right]+\left[2x\right]+\left[3x\right]+\dots +\left[nx\right]}{n^2}}$
where ${\displaystyle \left[x\right]}$ is the largest integer less than or equal to x. Prove that ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=\frac{x}{2}}$

Given series:
$\sum _{k=3}^{\mathrm{\infty }}[\frac{1}{k}-\frac{1}{k+1}]$
does this series converge or diverge?
If the series converges, find the sum of the series.

The terms of a series are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning.
$\sum _{n=1}^{\mathrm{\infty }}{a}_n$
$a_1=\frac{1}{5},a_{n+1}=\frac{\cos n+1}{n}{a}_n$