Round off (beautiful...) « Next Oldest | Next Newest »

 ▼ Tizedes Csaba Member Posts: 93 Threads: 15 Joined: Jan 1970 08-13-2003, 01:40 AM In the first math exam we got this example: What is the limit of this serie: (the calculator use not was allowable!) an=(1+1/(2^n+1))^(2^(n+1)) where n->infinity (n is integer) the limit is e^2 (~7.39), but look, what happen if we try to calculate it! Grab your 48, and plot it: 0 40 XRNG 0 10 YRNG 'n' INDEP ERASE DRAW You'll see, the value of function about 37.5-37.8 will be 1. (not 7.39) That happen, when 1+1/(2^n+1)=1 for 12 digit, the limit will be 1. Where? Its simple calculable: 1/(2^n+1)<5e-12 -> n>37.54 Csaba ▼ hugh Member Posts: 189 Threads: 23 Joined: Jan 1970 08-13-2003, 03:28 PM like you say, let k=2^n+1, then your problem is written (((1+1/k)^k)/(1+1/k))^2 ~ e^2 as k->inf. the difficulty for the calculator is (1+x/k)^k -> exp(x) as k->inf because you have here a classic 1+epsilon problem. the 1+eps problem is a defect of floating point numbers as real approximations. floats (inc calculator BCD) are only dense around zero. with a single floating point, 1+x-1 = x, only where abs(x) > 10^(-d) where d is the number of internal working digits. when IEEE floating point was invented (1983??), it was a big step forward in standardisation. i believe its time for calculator makers to move forward again onto a fundamentally new way of approximating reals.

 Possibly Related Threads... Thread Author Replies Views Last Post Received a beautiful HP-12C Prestige Kees van der Sanden 11 2,080 04-23-2012, 03:48 PM Last Post: Lode Vintage logarithm round up McAllan 9 1,727 03-18-2012, 11:17 AM Last Post: Dieter HP15C LE Keyboard Survey Take 3 (last round) M. Joury 33 5,197 09-28-2011, 02:28 PM Last Post: Michael de Estrada Such a beautiful sight Eric Rechlin 16 2,880 09-16-2011, 09:34 PM Last Post: Eddie W. Shore A beautiful HP-70 Michael de Estrada 7 1,573 03-09-2011, 12:02 PM Last Post: Juergen Keller HP 34C - which way round are the batterys fitted? Jon Duke 6 1,448 03-30-2010, 11:39 AM Last Post: Michael de Estrada Square or Round decimal point on the 50g Timespace 4 954 09-08-2006, 04:49 AM Last Post: Antonio Maschio (Italy) HP replaced 2 dead 49G+ with 2 beautiful 50G Iqbal 0 460 08-07-2006, 10:48 AM Last Post: Iqbal HP-86/87 Service ROMs - whip round request John 1 606 04-12-2006, 10:22 PM Last Post: Howard Owen Round hp invent calculator Charlie O. 5 1,026 03-31-2006, 03:46 PM Last Post: bill platt

Forum Jump: