La plataforma de Yahoo Respuestas cerrará definitivamente el próximo 4 de mayo de 2021 y solo estará disponible en modo de lectura. Ningún otro producto de Yahoo ni tu cuenta personal se verán afectados por este cambio. Si necesitás más información sobre el cierre de Yahoo Respuestas y cómo descargar tus datos podés acceder a esta página de ayuda.
Nested radicals?
The nested radical
a = {1 + [1+ (1.............) ^1/2 ]^1/2 + ..} ^ 1/2 = aureo number
Find the result if the series 1,1,1,1,1 is replaced by the series 1, 2, 3, 4, 5 ........
1 respuesta
- MorewoodLv 7hace 2 años
The first number: x = {1+[1+(1 ... )^(1/2)]^(1/2)}^(1/2) satisfies x²=1+x which, given x>0, allows only x=(1+√5)/2, the golden ratio. Of course one has to show convergence, but that is not too hard, or you can appeal to Herschfeld’s convergence theorem (look it up).
The second number {1+[2+(3 ... )^(1/2)]^(1/2)}^(1/2) is more interesting. It is commonly known as the Nested Radical Constant, although Herschfeld called it the Kasner number. A spreadsheet or calculator quickly gives a good approximation. I get about 3 digits for every 4 terms. However, according to the Online Encyclopedia of Integer Sequences http://oeis.org/A072449 "No closed-form expression is known for this constant". I haven't been able to improve on that but if anyone here can, please speak up and be famous!
