I am having trouble evaluating the sum [tex]\sum_{i=1}^{\infty}\frac{i}{4^i}[/tex] by hand.(adsbygoogle = window.adsbygoogle || []).push({});

My TI-89 is giving me an answer of 4/9 or 0.44 repeating, but I am uncertain how to go about solving this by hand and proving the calculator's result. To my knowledge, no identity or easy quick fix like the Integral test exists. I tried solving for the sum S by messing around with the first 5 or 6 terms, but it was all to no avail. Can anyone here point me in the direction towards evaluating this by hand in terms of a method(not asking anyone to do the problem itself for me)?

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# Evaluating infinite series

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