# 2017-13 Infinite series with recurrence relation

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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# 2017-11 Infinite series

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

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# 2016-11 Infinite series

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.)

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# 2011-20 Double infinite series

For a real number x, let d(x)=minn: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 + . . .

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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.)