how did Euler get from infinite series to pi?
828183
To find a closed form for that infinite sum was known as the Basel Problem. The sum is in fact zeta(2) where zeta is the Riemann Zeta function.\n\nBut how did Euler do it? Well, he was a very clever man. I suggest taking a look at this link for a detailed answer.\n\nhttp://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html
313965
This approaches infinity as n approaches infinity.  To demonstrate, I'll show you something even smaller that sums to infinity.\n\n1 + 1/2 + 1/4 + 1/4 + 1/8 [4 times] + 1/16 [8 times]...\n\nThis is equal to 1 + 1/2 + 1/2 + 1/2 + 1/2.....which "goes to infinity.  \n\nSince each term my this series is equal to or smaller then your series (1/4 + 1/4 < 1/3 + 1/4, and 1/8 + 1/8 + 1/8 + 1/8 < 1/5 + 1/6 + 1/7 + 1/8, and so on), and my series approches infinity, your series must approach infinity, too!
207196
There is an infinite number of primes.  They'll never be done.\n\nHow Many Primes are There? \n\nWe will now consider the question of how many prime numbers there are.  The following investigation of this question is over two thousand years old, and is attributed to Euclid. \n\nTwo possible answers to the question are that there are either finitely many primes, or there are infinitely many of them.  We will show that the first answer is not possible, and therefore there are infinitely many primes.  (This approach is referred to as "proof by contradiction."  In general, proof by contradiction works like this: we know that there are only two possible answers to a question; we assume that the first answer is the correct one, and then discover that this leads to an inconsistency; and so we conclude that it is actually the second of the two answers that is correct.) \n\nEuclid's Proof \n\nSo let's suppose that there are only finitely many primes, say n of them.  We could then make a list of these n primes, say p1, p2, ..., pn.  Then consider the integer \n\n\nobtained by multiplying all of the primes, and then adding 1.  This new number m is either prime or it is not.  If it is not, then it is divisible by some prime.  But we are supposing that the only primes are p1 through pn--and none of these primes divides m, for when m is divided by one of these primes, the remainder is 1.  This contradicts the known fact that every positive integer greater than 1 is divisible (evenly) by some prime.  So the conclusion must be that m is prime.  But clearly m is greater than all of the primes p1, p2, ..., pn , so this contradicts the supposition that p1, p2, ..., pn is a complete list of all primes.  Our conclusion is that there are not finitely many primes; there are infinitely many.
416833
There a number of different formulas for generating approximations to pi, most of which occur as infinitely long sums whose summation gives the exact value of pi, but whose sum out to the nth position approximates the value to pi to arbitrary precision. So, for most of these formulas, what you do in practice is perform the summation on a sufficiently large number of terms to get you as close as you want to the actual value of pi. Here's a web site that gives a number of formulas for approximating pi (I would write them here, but it's very difficult to write out equations on Yahoo answers):\n\nhttp://personal.bgsu.edu/~carother/pi/Pi5.html\n\nArchimedes was one of the first mathematicians to ever derive a formula to approximate pi... Here's an excellent web site which explains the method he used:\n\nhttp://personal.bgsu.edu/~carother/pi/Pi3a.html
1051417
Hi:\n\nRead the question by dreamchost1205 about the number pi and read iroc70 's answer it should explain thing very well.\n\nGood luck and best wishes to you
300967
There is no complete number. Pi just goes on and on and on. The best I could approximation I can give you is 3.14 or 22/7
166362
The number that you have here is only a good approximation to the actual value of PI. Others exist that are even better. Their common feature is that unlike PI itself, they are all rational, which is why each has an easy pattern, like the one you noticed just now, for this particular approximation. No easy pattern can be found for PI however, because, as has been said in so many answers, PI is irrational That is not enough though, sqrt(2) is also irrational and yet it has a clearly noticeable pattern, through continued fractions. To add to PI's misery however, it is also transcedental(sp?) and that probably explains why it is so hard to deal with PI, or with E, for that matter. Both are irrational and transcendents and in that respect are different from many other irrational numbers.\nHope that helps.
451479
hav u ever heard of the Principle of Mthematical Induction ?\nthat could solve ur problem in a jiffy!!\n\nHowever if u havn't, dont worry!!\n\nNote that, the sum in ur series(i.e. 1+1/2+1/3+...+1/n),where n>=2, is always less than 2, but greater than 1. so it can't b an integer.\nwanna know why?\ntry cramming up some theorems about Sequence & series of real numbers!! its a regulation part of any under-graduate level Real analysis course.
817918
This is an oldie:  What's i^i, where i = sqrt(-1)?   Ans:  By Euler's formula, we know that e^pi i = -1.  Therefore e^(1/2) pi i = i, or e^-(1/2) pi = i^i = 0.20788.
131038
Something is screwy with your question.  The second part is just a transformation of Euler's identity, e^ix = cos(x) + i sin(x)\n\nSo, your second expression simplifies to [e^(i 3pi/2)]/8\n\nIf I assume, like the previous poster, that your first expression should be (1+1/2i), then, just doing the algebra the left side of your equation gives, (1+1/2i)^6 = -117/64 - 11 i/16.\n\nFor the right side of your equation, for real, 1/8 cos 3pi/2 = 0\nFor imaginary, 1/8 i sin 3pi/2 = - i/8\n\nNow comparing left and right real and imaginary parts:\n0 != -117/64 and\n-i/8 != - 11 i/16\n\nSo, they are not equal.
48126
Well, there is no really simple answer. The most common is "Euler's proof," but even that is a bit cloudy. The fact remains that nearly every infinite series, even if it can be proven convergent, has an incalculable sum. That's not to say a computer can't do it, but then again, the longer the computer cranks, the more accurate the answer is. It's really not a solution when a computer calculates an answer--more like a limit that approaches the correct answer.\n\nHowever, there is hope! I have compiled some sources I found online to help you. Here's a brief description of the links:\n\n1. A discussion from Wolfram's Dr. Math about the general incalculability of infinite series, which specifically cites your problem as a counterexample (i.e. one that can be calculated).\n\n2. A forum posting about different approaches to this problem.\n\n3. A wicked cool collection of 14 proofs for your problem in PDF form. Download it! Print it! Use it to line your parrot cage when you're done!
634042
Theres a poster in my maths class that says "a trillion steps towards an infinitae pi" So i assume the best so far is a trillion number after the decimal point. WOW!
203743
SpawnOfSpock, the fact that there are an infinite amount of numbers does not imply there are also an infinite amount of primes. It is a serious enough question to have been proven in myriad ways by some of the greatest mathematicians.\nThe following is a common one. Suppose there were a finite amount of primes. Then the product p of all these primes is divisible by each prime. Consider the number p+1. Since p+1 was not in the set, it must be divisible by a prime number. Let that prime be s. But (p+1)/s = q + 1/s for some integer q and thus p+1 must also be prime. Thus our assumption that the set of all primes is finite must be wrong.
421226
Ha!  I just happen to know pi to 60 decimal places off the top of my head.  \n3.141592653589793238462643383279502884197169399375105820974944
401854
not sure--The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). http://en.wikipedia.org/wiki/Pi still not sure-2002 was 4 years ago there maybe a larger number. http://directory.google.com/Top/Science/Math/Recreations/Specific_Numbers/Pi/Digits/
797343
Pi is an infinite number. That means that it goes on forever. No one can calculate all of it. Currently, the computers can calculate only a small fraction of it.
1025141
The series is very simple.\n\n0  1  1  2  3  5  8  13  21  34  55  89  144  ... \n\nYou just add the two previous numbers to the next number.\n\nThere are tons of websites related to your question, so you just haven't looked, and so don't know enough to ask a specific question.