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}}.

\]

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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 a_{1}, a_{2}, … 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 a_{1}>0.)