Let \(a_0 = a_1 =1\) and \(a_n = n a_{n-1} + (n-1) a_{n-2}\) for \(n \geq 2\). Find the value of
\[
\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.
\]

Find the value of
\[
\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.
\]

For a positive integer \( n \), define \( f(n) \) by
\[
f(n) =
\begin{cases}
0 & \text{ if } n \equiv 0 \pmod{5} \\
1 & \text{ if } n \equiv \pm 1 \pmod{5} \\
-1 & \text{ if } n \equiv \pm 2 \pmod{5}
\end{cases}.
\]
Compute the infinite series
\[
\sum_{n=1}^{\infty} \frac{f(n)}{n} = 1 – \frac{1}{2} – \frac{1}{3} + \frac{1}{4} + \frac{1}{6} – \dots.
\]

(This is the last problem of this semester. Thank you.)

For a real number x, let d(x)=min_n:integer (x-n)². Evaluate the following double infinite series:
. . . + 8 d(x/8)+4 d(x/4) + 2 d(x/2) + d(x)  + d(2x) / 2 + d(4x)/4 + d(8x)/8 + . . .

Let a₁, a₂, … be a sequence of non-negative real numbers less than or equal to 1. Let \(S_n=\sum_{i=1}^n a_i\) and \(T_n=\sum_{i=1}^n S_i\). Prove or disprove that \(\sum_{n=1}^\infty a_n/T_n\) converges. (Assume a₁>0.)