Question

how pi is evaluated ?

Answer 1

pi=4/1-4/3+4/5-4/7+4/9-4/11.............

Answer 2

i know only the definition of pi , pls explain how that series is obtained

Answer 3

Good question! There are numerous ways. The following is one. When you get to calculus, you will find the following.\[\text{arcsin}(x)=\int \frac{dx}{\sqrt{1-x^2}}=\int\left(1+\frac{x^2}{2}+\frac{3x^4}{8}+\cdots\right)dx=x+\frac{x^3}{3}+\frac{3x^5}{32}+\cdots\]Now, recall that \(\text{arcsin}(1)=\frac{\pi}{2}\). Thus, we can say the following.\[\text{arcsin}(1)=\frac \pi 2 = 1+\frac{1}{3}+\frac{3}{32}+\cdots\]\[\boxed{\displaystyle \pi=\left(1+\frac{1}{3}+\frac{3}{32}+\cdots\right)}\]The series cliumay9 cited uses the same method except with arctangent. It's easier to do, but the series converges very slowly (to get a degree of accuracy to the second decimal place, you need about 50 terms!).

Answer 4

Whoops! I meant \[\boxed{\displaystyle \pi=2\left(1+\frac{1}{3}+\frac{3}{32}+\cdots\right)}\]

Answer 5

actually pi is the ratio of the circumference and radius/diameter(pls correct me, IDK whether it is r or d) so our ancestors cleverly found the ratio between them by using 3-sided polygon to n-sided polygon..... thats how they explain pu

Answer 6

its d , the diameter

Answer 7

And that should actually be a 40 in the denominator because I did the integral slightly wrong...hmm I'm a little off today. haha But the idea is there!\[\boxed{\displaystyle \pi=2\left(1+\frac{1}{3}+\frac{3}{40}+\cdots\right)}\]

Answer 8

thank u...

Answer 9

cliumay9, have you tried actually doing that method yourself? Doing so requires trig functions, which in turn requires knowledge of the value of \(\pi\). You'd otherwise have to do a manual summation for a large polygon which is very inefficent, and won't get you the accuracy of \(\pi\) that we have today.

Answer 10

thank you yak