(((1/2))/(1+(1/2)))+(((1/3))/((1+(1/2))(1+(1/3)))) +(((1/4))/((1+(1/2))(1+(1/3))(1+(1/4))))+…+(((1/1999) )/((1+(1/2))(1+(1/3))...(1+(1/1999)))).
(((1/2))/(1+(1/2)))+(((1/3))/((1+(1/2))(1+(1/3))))+(((1/4))/((1+(1/2))(1+(1/3))(1+(1/4))))+…+(((1/1999))/((1+(1/2))(1+(1/3))…(1+(1/1999))))．
<gadget id="calculator">1 / 2</gadget>
<output>1/2 = around 0.5</output>

<gadget id="calculator">1 / 2_000</gadget>
<output>1/2_000 = around 0.0005</output>

<gadget id="calculator">(1/2) - (1/2_000)</gadget>
<output>999/2_000 = around 0.4995</output>

<gadget id="calculator">(999/2_000) * 2</gadget>
<output>999/1_000 = around 0.999</output>

<result>999/1_000 = around 0.999</result>