The sum of infinite numbers

Two weeks ago, at Metroplex Math Circle presentation "Primes and Composite Numbers", the presenter used a formula for the sum of the squares of numbers from 1 to n in his solution of a problem. I thought that I might be able to derive the formula.

1^2+2^2+ .......+n^2= n*(n+1)*(2n+1)/6

To derive the formula turns out to be not trivial, but to prove it was not hard, using induction. During my attempt to derive the formula, I tried converting the problem into a recursive procedure, into a continuous problem instead of discrete problem so that I could use calculus....., all without success (1). The idea of using continuous function for solving discrete number problem stays with me.

The next day, I encountered another formula for the calculation of Pi

Pi = 4 ( 1/1 - 1/3+1/5-1/7+1/9-1/11+1/13 ......+1/(2n-1) - 1/(2n+1) + ...........)


The irrational Pi can be determined by this infinite series of rational numbers - fractions!!

The sum of infinite numbers

My typical approach to find the sum of infinite numbers is to find a formula for the sum of first n terms of the series, and then take a limit (n--> infinite). For example

1/(1*2) + 1/(2*3)+1/(3*4) + ......+ 1/(n*(n+1))+ ......

Let S be the sum of the infinite series, Sn be the sum of the first n terms

Note that 1/(n*(n+1)) = 1/n - 1/(n+1), so

Sn = 1/1 -1/2+1/2-1/3 +1/3-1/4 + ...1/n -1/(n+1) = 1-1/(n+1)

S= Sn(N--> infinite) = 1

This approach is apparently not suitable for the calculation of the formula for Pi listed above

Taylor Expansion for sum of infinite series

Looking at the formula

Pi = 4 ( 1/1 - 1/3+1/5-1/7+1/9-1/11+1/13 ......+1/(2n-1) - 1/(2n+1) + ...........)

the idea of using continuous function to calculate the discrete came back to me. This formula seems to relate to trigonometric functions, arcsin or arctan. Checking handbook of math, I found this formulation of Taylor expansion of arctan (x) when -1 <= x <=1)

arctan(x) = x- x^3/3+x^5/5-x^7/7 +x^9/9 .....+(-1)^n x^(2n+1)/(2n+1) + .....

Take x=1, the above Taylor expansion yield

Pi = 4 ( 1/1 - 1/3+1/5-1/7+1/9-1/11+1/13 ......+1/(2n-1) - 1/(2n+1) + ...........)

With Taylor Expansion of various functions, one can easily determine many seemly impossible sum of infinite series .



(1) Upon literature search, it turned out that more general solution exists, and the sum of powers (fixed exponent m) is always a polynomial of n of degree m+1. I will find the derivation of it and summarize it in a future post.

Fighting for Auto-Insurance Coverage - a story

While filing stacked files on top of our file cabinet into proper place this Sunday, I came across this bulging FedEx envelope - which contained all the correspondence on a disputed auto insurance coverage. It is a story about Lily's fight to protect her rights.

The accident

A Monday morning last November, Lily called me on her way to work - she had an accident! Her car was damaged but she was not hurt. Here is her description of what happened to the other driver's insurance company.

"In the morning of November xx, 2010, around 8;25am, I was driving on "A" street heading south to work. This portion of the road is a 3 lane one-way street. I was in the left lane. As I was going straight on "A" road, about to cross the intersection of "A" street and "B" Boulevard, a car in the center lane, driven by Ms. M, made a left turn in front of me. That car hit the right front of my car. I stopped at the intersection, while she kept driving on "B" after the collision. But she was stopped by a traffic light after her turn. So I had to make a left turn to catch Ms. M. Ms. M refused to provide insurance information so I called police. Police and a Fire Rescue cars arrived at scene. Police report was filed"

At the time, the police asked both drivers drove their cars to a nearby parking lot and questioned both drivers separately. Lily also asked police to check on the traffic signs on the road - left lane where she was can go straight or turn left; center lane has to go straight (as shown in the illustration).

Simply put, the other driver made a left turn from middle lane while Lily was driving straight from left lane.

The claim

It turned out that Ms. M's insurance company is a small company we have never heard of before , and the company was already debunked in Oklahoma. Lily was distressed, and I tried to comfort her - our insurance company will take care of everything.

Our insurance filed claim for us to the other company through a process called subrogation, and also recommended Lily to let their insurance inspector to check her car and then go to fix the car paying our deductible first. She followed these suggestions. What happened next was unexpected

The other driver, Ms. M., changed her story while responding to the insurance company inquiry. She admitted that she made an illegal turn, but she said that’s not where the accident happened, instead she and Lily both turned at the intersection and Lily moved into her lane after the turn and hitting her car. The damages on the car can’t tell whose story is true, as we were told by both insurance companies. We had to wait for police report to clear things up.

The police report was consistent with Lily’s story. This cleared claim with our insurance company. Ms M.’s insurance denied Lily’s claim, ignoring the police report. The reason they gave to Lily was that "There’s no witness therefore they chose to believe their client and denied Lily’s claim" - ignoring the fact that Ms.M changed her story from what she told police at the scene.

Lily called the adjuster of Ms. M's insurance, and she was sympathetic, and asked Lily to send additional information. After Lily sent additional information, basically a written description of the accident, plus police report, she received another letter of denial of liability (March 2011). She called the adjuster's number at the insurance company, and was told the adjuster was not with the company anymore, and a manger took over the case. She was told the decision was final.

Lily called our insurance company about the denial of liability and was told that the money involved was too small to start a legal fight over this denial of liability. Our insurance company would not do anything!! She was outraged with both companies.

Lily could not let the injustice go.

How could people just get away with lying, how come nobody cares about the truth and fairness? Lily filed complaint with Texas Department of Insurance, she wrote a letter to the CEO of Ms. M insurance company, which was found on the company's website. After these she’s hoping to put a closure on this case.

Late April 2011, Lily received a check from the other insurance company unexpectedly. When she informed our insurance company, the agent who handled the case had a hard time believing it.

In May 2001, Texas Department of Insurance replied to Lily's complain, and relayed the other company's decision to us " .....this claim was reopened, and due to the fact that we never received (our) insured driver's written statement (regarding this accident), the Company decided to accept the complainant's version of loss. ......". It was almost exactly six months after the accident when Lily finally had a closure on this case.

Lessons Learned

First thing first, get a witness when you can, it’s worth a million words.

In cases like this, or even more serious litigation, seek help from friends, colleagues and experts to find a way forward.

In the process of arguments, being impassioned is always better than being emotional.

Our country is a country with law and order ; even though justice sometimes requires time and resources to be upheld.

Don't give up easily on justice, fight for it.

Random thoughts on electronic gadgets and etc

With the passing of Steve Jobs last week, many people expressed their appreciation, admiration profusely. Most people relate to him via iPod, iPhone, iPad, iTouch .....the gadgets he personified. There is no doubt that he was an exceptionally talented technologist and a superb marketer.

I looked at his impact on the society from a different angle.

Was iPod, iPhone, iPad .... revolutionary?

Was iPhone and etc revolutionary? The answer is NO, technology wise. The iPhone, iPad ... and series are basically repackaging of existing technology. The genius in this series of Apple products is packaging and marketing.

Some commentators placed Steve Jobs historically at the summit of technology along with Edison, Ford. But most were careful, and rightfully so, not to attribute it to his work on the development and marketing of the gadgets, but to his pioneer work in the development of personal computers, with Steve Wozniak.

It is important to note that most literature attribute the inventor of PC to Henry Edward Robert.

The impacts of electronic gadgets and internet

The impact of i-series Apple products, along with other gadgets, and advances in internet on human is historical. They make personal communication, personnel entertainment so much more convenient, which can be achieved essentially any time, any where.

The negative impacts of electronic gadgets and internet are significant as well.

First and foremost is the negative impact on person to person interaction. On one hand it facilitates personal communications, on the other hand, it significantly reduces person to person interaction, especially in the young. Many people text each other more than they talk to each other.

It facilitates distractions from real work - be it being work work, or study for students. It has fostered a generation of young people who can not concentrate on real work, who do not know how to separate leisure from work or study.

Politically, it helps to make the vocal minority louder than ever before.

Is electronic gadget industry GREEN?

One tribute to Steve Jobs was that he demanded perfection in making iPod, iPhone ...instead of using plastic cases for the devices, he insisted on using Aluminum, which made i-series device shiny, slick and fashionable. There is no problem to it if the purpose is to make the devices to last.

From sustainability point of view, the issue with Apple and electronic gadget industry in general is that they encourage people to ditch their perfectly functional gadgets for newer, more fashionable new generation of devices. Unfortunately many young people are addicted to the newest and the gadget industry has its way. Even worse, for those who want to be frugal and fashionable at the same time, they could not hang on to the older devices because there are new functions that are purposely developed not to play on older devices! Worst of all, it is cheaper to buy new gadgets than to repair them. One such practice is highlighted by a headline like this " Apple's pentlobe screws for iPhone take "planed obsolescence to the next level".

Of course it is not just gadget industry, it is the electronic industry, and consumer product industry in general. “E-waste is now the fastest growing component of the municipal solid waste stream because people are upgrading their mobile phones, computers, televisions, audio equipment and printers more frequently than ever before. Mobile phones and computers are causing the biggest problem because they are replaced most often.”

This leads to million tons electronic waste every year. In the process of depleting natural resources, the e-waste also pose threat to the environment and thus potentially harm human life indirectly. In this sense, Apple's i-series devices and their marketing practice helped to amplify this waste.

I am not a fan of electronic gadgets and I resisted purchasing such devices until I saw its practical use for my family. We did not have a cellphone until 2000 and we don't upgrade our cellphones until they are not functional any more. We used our first home computer for nearly 10 years until we had problems with its operating system which Microsoft no long supported. We did not buy any Apple's i-series device until last year - we purchased an iTouch for our teenage son. All functions iTouch provides are available on our second home computer, the convenience it provides, also bring tremendous distractions to him from study and sleep, and headaches to me.

Steve Jobs

His commencement speech at Stanford was replayed many times on network news. It was a great reflection on life. I found its full text at Stanford's website. I read it and I appreciate it more. I excerpt a few paragraphs from it.

"You've got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven't found it yet, keep looking. Don't settle. As with all matters of the heart, you'll know when you find it. And, like any great relationship, it just gets better and better as the years roll on." I can not agree more because I have a similar approach to work and life.

He also said: " .... Your time is limited, so don't waste it living someone else's life. Don't be trapped by dogma — which is living with the results of other people's thinking. Don't let the noise of others' opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition." So true!! But the generation iPod, iPhone, iTouch .. fostered is exactly the opposite what he pitched. They follow the crowd, they want the newest gadgets because they are new, in fashion and everyone else has it. I don't blame him for it - it is consumerism that our economy is based on, it is the consumerism that drives all business.

Steve Jobs, an exceptionally talented technologist and a superb marketer, a person followed his own heart in life, rest in peace.

Gravity at Work - "Space Force" - a Toy

"Space Force" was a toy we bought 4 or 5 years ago when we visited Houston Space Center and Science Museum. We placed it on top of the bookshelf along with a model of space station as decoration to our study. From time to time I did wonder that how it could be played - but we lost the box and instruction, so it has stayed on top of the bookshelf all these years.


A weekend several weeks ago, Nick was asked to stay away from computer for an hour to rest his eyes. He wandered around the house, nothing really interested him that morning, so he came back to the study (where the computer is), then he saw the pendulum like toy on top of the bookshelf - he asked me to take it down for him.

How to play

He took the toy to playroom and tried to figure out how to play it. He was quiet and stayed there for an hour or so. Then he declared to me that he knew how to play it.

"You place it on the floor, then place the steel ball at the pinned side of the wires. You push the wires apart slowly and try to make the ball roll as far as possible" . "You see, there are numbers on it:

Mercury -250
Earth 250
Mars 500
Jupiter 1000
Saturn 2000
Pluto 5000

You score points when the ball drop into a hole."



Trying to make the ball roll to the Pluto - the furthest planet and highest score hole, he played it again and again. He was fascinated by it. When he finally figured it out, he asked to have a game with his Mom and easily beat her.

After lunch, he kept playing on. I was curious now - I sat by him, observing him play.

A game of gravity and inertia

The top, or the narrow side of the toy, is lower than the wider side or bottom of the toy, when it is placed horizontally on the floor or a table. When the two wires touching each other at the bottom side the ball will stay at top side.

To make the ball roll to the direction of bottom, one has to push the wires apart - make the ball to fall in between wires, so gravity will make the ball roll and gain speed. If the wires are left apart, eventually the gap (which changes with distance) between them is too large the ball will fall.

The trick: as the ball gaining speed, pushing the wires close, now the ball has to go from low to high, the ball is moving against gravity! No worry, inertia can keep the ball moving forward for a second or so if the speed is high enough. Just as the ball's speed decreases to zero, push the wire open - the gravity will be a favorable force again - pushing the ball toward Pluto!

The competitions

Nick played a few times by himself, and boasted to me that he beat his Mom easily. "Let's play the game against each other" He challenged me. "OK", I said, "But I need to practice a few times". He agreed.

We had a pencil and a piece of paper ready before we start the game. We would each play 10 times, and the one has higher total score win. Nick played first, he got to Pluto once, but he also dropped the ball at Mercury once - negative points. I reached Pluto twice and the rest were all further away from earth. I won the first round.

Nick did not whine, and was in a good mood. After listening to my explanation of the mechanics of the game - gravity, inertia and etc, he asked to have a rematch.

Playing the game requires more than understanding of the mechanics, when one knows the mechanics - by knowledge or by trial and error, it is more of an art - how fast or slow to push the wires apart, when to push the wires to each other, and when to push them apart again. The second time, Nick reached the Pluto 3 times, I got there only once. Nick won the rematch and was very pleased.

This is an educational and fun toy.

My name is Asher Lev, a good read

- by Lily

I got introduced to this book by my son, who was reading it for English class. “It’s a really good book, I’m glad we are reading it in class,” He told me one night. Out of curiosity, I picked up a copy from the library and found myself drawn to it.

The story is set in a Jewish community in Brooklyn, New York around 1950s. This is a community with many pious Hasid Jews, who keep Kosher, pray three times a day, believe in the master of Universe, and despise art and science. This is where the main character Asher Lev grew up. Asher’s parents are both devoted Ladover Hasid, especially his father who traveled for the Rabbi around the world to establish Jewish community and schools. But Asher was born with a special talent, painting. The story is about how Asher struggled growing up in such an environment with such a talent, torn between his talent, his family, and his religion.

I don’t understand the Jewish religion, and I don’t know much about art. But I am drawn to this book. Of course, the author, Chaim Potok, was a great writer. His exquisite description of how a boy perceives his world while growing up is very touching. Asher’s uncontrollable habit of drawing and subconscious disclosure of his thoughts through his drawing, Asher’s perception of his parents, the mingled feelings of love, longing, anger, and refusal between them, the wise Rabbi, Asher’s unorthodox teacher who not only taught him how to draw but also how to be a true artist, the book captured all of those moments and is filled with true feelings.

More importantly, the issues the author raised in the book go way beyond any particular religion or a specific talent. I can’t help pondering if my definition of “useful” and “useless” skills is hindering the development of my kids’ “hidden” talent, how do you balance everyone’s need in a family without losing yourself, what is the best way to guide a person through life so he can make the right decisions and be true to himself, wouldn’t the world be much more enjoyable if only we can be more open-minded and appreciate each other just a little more?

I feel compelled to thank my son’s English teacher for introducing to us such a wonderful book, and she told me there is a sequel to the book, “The Gift of Asher”. Right, you know what I’m reading now :)