+4
Compute
\(\[1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.\]\)
Partial sums when n=10
0.5, 0.5, 0.375, 0.25, 0.15625, 0.09375, 0.05468 75, 0.03125, 0.01757 8125, 0.00976 5625=1.98828......etc.
When n = 25
0.5, 0.5, 0.375, 0.25, 0.15625, 0.09375, 0.05468 75, 0.03125, 0.01757 8125, 0.00976 5625, 0.00537 10937 5, 0.00292 96875, 0.00158 69140 63, 0.00085 44921 875, 0.00045 77636 719, 0.00024 41406 25, 0.00012 96997 07, 0.00006 86645 5078, 0.00003 62396 2402, 0.00001 90734 8633, 0.00001 00135 8032, 0.00000 52452 0874, 0.00000 27418 1366, 0.00000 14305 11475, 0.00000 07450 58059 7=1.999999......etc.
∑[(n*1/(2^n)), n, 1, ∞] = 2
