Answer:
Method
Consider ∑n=0∞xn=11−x, where |x|<1." role="presentation">Consider ∑n=0∞xn=11−x, where |x|<1.Consider ∑n=0∞xn=11−x, where |x|<1.
Then differentiating both sides w.r.t. x, we have
∑n=1∞nxn−1=1(1−x)2" role="presentation">∑n=1∞nxn−1=1(1−x)2∑n=1∞nxn−1=1(1−x)2
Putting x=12" role="presentation">x=12x=12 gives
∑n=1∞n2n−1=1(1−12)2=4" role="presentation">∑n=1∞n2n−1=1(1−12)2=4∑n=1∞n2n−1=1(1−12)2=4
12∑n=1∞n2n−1=2" role="presentation">12∑n=1∞n2n−1=212∑n=1∞n2n−1=2
∑n=1∞n2n=2" role="presentation">