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CommentsThe Pirate Bay, the highly-controversial BitTorrent site that’s widely believed to be the major destination for illegal content on the Internet, has a problem: it needs money to stay in business.

If the site is shut down by a court order, it won’t just go down the drain: its servers will be seized and auctioned off. The three founders of the site have already paid out a significant amount in order to secure its hosting, which they have done on a commercial basis since 2006. But the site is barely making any money at all from ads, even though some of the largest web companies — including Google, Yahoo, and Facebook — pay to advertise there.

According to Richard Chomas, one of the site’s founders, the company makes “about 60” cents per month for each visitor — and that’s if it’s lucky. Chomas’ estimate is based on a rough calculation of the site’s share of traffic. According to Chomas, the site only receives 1,000 to 2,000 unique visitors a day, which, he estimates, makes the site’s cash about $1.50 a user.

That’s a tidy amount compared with the ad-supported web sites that Chomas estimates are out-earning the Pirate Bay. However, for a site that represents “the average person’s online habits,” it’s not very much.

For example, Chomas says the Huffington Post, an online newspaper that recently sued Google in the hopes of getting more traffic, makes $30 for each user — which is significantly more than the $1.50 the Pirate Bay makes for each visitor.

The Pirate Bay says it can’t afford to pay its lawyers

And it’s not just the amount of money the Pirate Bay is making that’s alarming: The site says it can’t afford to pay its lawyers to represent it in a lawsuit against a Swedish court ordered to shut the site down. “We have to lay off our legal team,” one of the site’s founders, Fredrik Neij, told a friend who contacted him.

The claim that the site has fallen on hard times is supported by the fact that the site doesn’t even have a bank account, relying instead on wire transfers to make purchases. And while it does technically serve as a content distribution network, it’s not the one people imagine when they think of the word “pirate.”

“We’re not pirates,”

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Proving $\lim_{k\to\infty}\frac{1}{k+2}\sum_{i=1}^k\frac{k-i+1}{(k-i+1)!}=1$

How can I prove that
$$\lim_{k\to\infty}\frac{1}{k+2}\sum_{i=1}^k\frac{k-i+1}{(k-i+1)!}=1$$

A:

Use the Euler-Maclaurin formula to prove that $$\sum_{i=0}^k \frac{1}{k+2-i}=\int_0^1 \frac{1}{1+x} \, dx$$
then change your sum to an integral as well.

If you wanted to be fancy, you could rewrite the integral as a power series and you might be able to use the Central Limit Theorem to your advantage.

A:

Using the exponential generating function, we get
$$\frac{1}{k+2}\sum_{i=1}^k \frac{k-i+1}{(k-i+1)!} = \frac{1}{k+2}\frac{d^k}{dx^k}\frac{\sinh(x)}{\sinh(x)} \bigg|_{x=0} = \frac{1}{k+2} e^{ -x}\frac{d^k}{dx^k} e^{x} \bigg|_{x=0} = e^{ –
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