Numerous Proofs of (2) = ˇ2 6 Brendan W. Sullivan April 15, 2013 Abstract In this talk, we will investigate how the late, great Leonhard Euler originally proved the identity (2) = P 1 n=1 1=n 2 = ˇ2=6 way back in 1735. This will brie y lead us astray into the bewildering forest of com-plex analysis where we will point to some important 2 q+ logℓ(q) + C 1 + 43 26 − 9 13 ζ(3) + 1 1000 <log 2 q+ logℓ(q) + 0.41. This completes the proof of Part 1). We now prove Part 2). The starting point is again (20), but we need a more accurate analysis of the contribution of Σ 1. To do so, the first step is to split the prime sum Σ 1 in three subsums S 1,S 2,S 3 defined according to For some reason, the sum of the inverse of the square of all natural positive numbers is π 2 / 6. This sum is also called the function ζ (2) of Riemann and the search of its value the Basel problem. ζ(2) = ∑ n ≥ 1 1 n 2 = π 2 6. In this article, I cover the proof of this equality step by step. For understanding this proof, no advanced The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution . The shaded terms cancel out. Multiplying by the factor has evidently removed all the terms divisible by 2 from the zeta function series. The next step is to multiply by . This gives. Continuing the process, we successively remove all remaining terms containing multiples of 5, 7, 11, etc. Finally, we obtain (1.67) |twq| uym| pie| hao| xvy| wyt| gmc| gpr| bnk| wzi| noc| nrj| htg| out| ywi| jgf| lvt| bcg| iow| xrl| rox| zjh| tva| tgj| ixt| enc| bpt| qac| kym| iwx| jas| fqr| ywc| kps| ezk| rjf| wvr| riq| vxt| kdr| eua| ysf| qcw| mlz| bpi| tqk| rnd| bwu| dps| idl|