Evaluate the following sum (with proof):
\[
\sum_{k=0}^{\infty} \frac{1}{(6k+1)(6k+2)(6k+3)(6k+4)(6k+5)(6k+6)}
\]

Evaluate the following:
\[
\frac{1}{1^2 \cdot 3^3 \cdot 5^2} – \frac{1}{3^2 \cdot 5^3 \cdot 7^2} + \frac{1}{5^2 \cdot 7^3 \cdot 9^2} – \dots
\]

Determine whether or not the following infinite series converges. \[ \sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.\]

For an integer \( p \), define
\[
f_p(n) = \sum_{k=1}^n k^p.
\]
Prove that
\[
\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.
\]

Let \(k\) be a positive integer. Let \(a_n=1\) if \(n\) is not a multiple of \(k+1\), and \(a_n=-k\) if \(n\) is a multiple of \(k+1\). Compute \[\sum_{n=1}^\infty \frac{a_n}{n}.\]

Determine all positive integers \(\ell\) such that \[ \sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}\] converges and if it converges, then compute its value.