Find a power series for pi.

Question

Using the derivative of arcsin(x) and the binomial theorem, derive an infinite power series for pi, and give the general term (nth term) for this series. Work out the first four terms of the series and use them to give a value for pi to 3 decimal places. Hint: Use the fact that sin((pi)/6)=1/2.

Comments

Hint:

sin(y)=x, derivative is cos(y)dy/dx=1, dy/dx=1/cos(y)=1/sqrt(1-x^2)=(1-x^2)^(-1/2).

y=integral((1-x^2)^(-1/2)dx), the integrand can be expanded binomially using the general formula (1+X)^n=1+nX+n(n+1)X^2/2+..., where X is replaced by -x^2 and n=-1/2. Each term can easily be integrated with respect to X: X+nX^2/2+n(n+1)X^3/6+..., making the substitution for X and n. The constant of integration, k, can be found by substituting (0,0). It turns out that k=0. y=f(x) where  f(x) is the result of integration and X is replaced by -x^2.

To find a series for (pi), use y=(pi)/6 and x=1/2 because sin((pi)/6)=1/2. So (pi)=6f(1/2). This is an example of a series for (pi).

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I am sorry. I cannot answer that. I don't even know what that means. I am in 6th grade. I think it is like 8th grade math or highschool math. Sorry Rod. I want to help, but I don't know it. You have been helping me, but I don't know how to repay you because some of these questions are a little advance for me. If you have a question, I will try to answer it for you soon as possible. By the way, I send a private message for you. Read it and answer it back to my profile. OK?

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No, I wouldn't expect a 6th grader to understand that because it's calculus, a special branch of math you wouldn't meet until you're a few years older and in high school. So don't worry, you don't owe me anything in the way of answers. But my reward is in seeng you better able to tackle questions as you understand and use what you already know.

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Rod, I think I found out how to do it! I looked it up online. I will show you the answer for it right now. :)

Answer 1

Start with

1/(1+w) = 1 - w + w² - w³ + ...

Now subsitute x² for w:

1/(1+x²) = 1 - x² + x⁴ - x⁶ + ...

Then integrate both sides (from x=0 to x=y):

arctan y = y - y³/3 + y⁵/5 - y⁷/7 +...

and plug in y=1, to get

Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

Voila!

There are other pi formulas that converge faster.

Presentation Suggestions:
An alternate way to present this is to start with the well-known formula for Pi, and then present this as a "justification".

The Math Behind the Fact:
Well, we glossed over the issue of why you can integrate the infinite series term by term, so if you wish to learn about this and more about Taylor series, this material is often covered in a fun course called real analysis.

 

Comments

Yes, you did well to find that solution. The one I suggested does converge faster, but top marks for discovering a solution!

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Rod, do you have any questions? I will try to answer them. Your advanced it math! I am so happy for you? How old are you because that your so good at math! I know you might be out of school a long time ago. Your really know your math very well, so as other people on here in this website.

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To you I must seem very old when I tell you I'm nearly 72, and it's a very long time since I was at school and what you would call college, but we call university. But when you really have a grasp on something like math, or some parts of it, you never lose it, because you've become so used to the ideas. There's a lot I don't know, but I like to look for and find an easy way to understand new ideas. That also helps me to explain to someone like you, little buddy, ideas that I fully understand, so that you can learn them, too.

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Rod, your sweet. :) To me, your not old. My grandmother is older than you. She is close to 80 years old.To me, she looks like my mothers age. Your avatar looks like your 42 years old or so. You have helped me with my homework so much! I can't get my school done at home because I am having trouble with math, science, social studies, and languge arts. I am about 11 years old. I am just a kid. I know some things about those subjects though.