## An interesting sequence and its limit

Problem: Consider a sequence of numbers $$a_m(j)$$ such that $$a_m(j+1) = a_m(j)^2 + 2 a_m(j)$$ for all $$j \geq 0$$ and $$m \geq 0$$ with $$a_m(0)=\frac{d}{2^m}$$ for some real number $$d$$. Determine $$\displaystyle\lim_{m \to +\infty}a_m(m)$$.

Solution:  The above recursive relation can be rewritten as follows:

$$1+a_m(j+1)=(1+a_m(j))^2$$

One of the most underrated tricks in solving recusive relations is to substitute in a new sequence whose pattern has a clear path to solution.  In this case the substitution glares at you. Let $$b_m(j)=1+a_m(j)$$.  The new sequence satisfies the recurrence

$$b_m(j+1)=b_m(j)^2$$

and has an initial condition $$b_m(0)=1+\frac{d}{2^m}$$.  We then can step through the first few cases:

$$b_m(1)=(1+\frac{d}{2^m})^2$$,

$$b_m(2)=(1+\frac{d}{2^m})^4$$,

until we get to the desired case:

$$b_m(m)=(1+\frac{d}{2^m})^{2^m}$$.

Using the known limiting relation $$\displaystyle\lim_{m \to +\infty}(1+\frac{z}{m})^m=\exp{z}$$, and the relation $$b_m(j)=1+a_m(j)$$, we get the desired result:

$$\displaystyle\lim_{m \to +\infty}a_m(m)=\exp{d}-1$$.

## Bullies and genocide

I plead with you to read this piece by Shalom Auslander.  I am ashamed to have thought of this myself.  Then again, no I’m not…

Brian is a fat dumpy turd who is going to get his ass kicked one day. Not by me, because I’m almost 40, and he’s not yet eight. But he’s a bully, and he’s been bullying my son, who is not yet five. I look at Brian—almost half my height and damn near double my weight, his barely-fitting XL “Transformers” t-shirt covered with bits of cake and ice cream, his fat little legs already starting to splay out in the manner of the morbidly obese, the cursed beams of his insufficient structure already too weak to cope with the oversized load they are being asked to support, his hollow, heavy-lidded eyes blinking out at the world in the sort of dumb, mouth-breathing incomprehension you see in mall kids and SS men and Glenn Beck—and I think about the genocide books I’ve been reading. They all wonder why. They all seem to think there’s a reason, and that if they can identify that reason, these horrible crimes will never happen again. The reason, they say, is poverty. The reason is racism, the West, the East, religion, atheism, capitalism, communism. But it isn’t.

The reason is Brian.

There is no reason for Brian. I’d like there to be. But there isn’t. Brian just is. Brian happens. Is Brian going to lead Hutus to slaughter Tutsis? I don’t know. Perhaps he’s not that ambitious. But if Brian were a Hutu, Brian would hack a Tutsi, no question about it. Brian would hack a lot of Tutsis. Brian would be the Hutu in that news footage, dancing around the mangled corpse of a young Tutsi with his bloody machete raised triumphantly overhead. Only fatter. And eating a Twinkie.

“That fat little asshole,” my wife said.

“Brian.”

She had just come upstairs from tucking our son into bed, which was when he told her what had happened. Brian had been teasing him on the bus, poking him and trying to steal his GI Joe doll.

“That fat little asshole,” she said again.

“Okay,” I said, putting down The History of Torture and Execution from Early Civilization Through Medieval Times to the Present. “Just calm down.”

My wife is Middle Eastern; if you don’t stop the rock-throwing right away, pretty soon you’re shutting down East Jerusalem. I reminded her that our son has a vivid imagination, and that while something probably did happen, we don’t know for certain exactly what it was, and after all, this is Woodstock, it’s not like he was attacked by the Crips, and eventually, by the way, he is going to have to learn to fight his own battles.

“Okay,” she said. “You’re right.”

My son began to cry. I went downstairs, sat on the edge of his bed, and asked him what was wrong.

“I was having a bad dream.”

That fat little asshole, I thought.

“We’re on the bus,” he said, “and he’s picking on me and stealing my toys and then the bus stops and it’s my turn to get off but he won’t let me and the bus leaves and I can never get home.”

That fat little asshole.

I wanted to tell him that he didn’t need to worry, that there was a man who lived a long time ago named Charles Darwin, and that Darwin figured out that we all evolved from monkeys and apes, and that some of us are more evolved, and some of us are less evolved, and some of us—the Brians of the world—have actually devolved somehow into something less than apes. But I heard my shrink in my head, telling me that all your children need to know is that you love them, and will always love them, and that’s all that matters. And so I told my son that I love him, and that I would always love him, and that was all that mattered. I may have mentioned something about the fact that if Brian ever touched him again, I would cut him up into tiny bits, stick them on skewers, put him on the grill until he was all cooked up, and then feed him to the dogs. And that I really, really love him.

My son laughed.

“Will you mash him up into peanut butter and put him on a sandwich?”

I laughed and said I would.

“Will you drop him off a building and drop a piano on his head.”

He’s been watching a lot of Bugs Bunny lately.

“Will you…”

“Okay, buddy, it’s time to get some sleep.”