Question 1209795
<br>
{{{1+1/2+1/10+1/20+1/100+1/200}}}+...<br>
Split the series into two purely geometric series:<br>
{{{1+1/10+1/100}}}+...<br>
and<br>
{{{1/2+1/20+1/200}}}+...<br>
The infinite sum of the first series is<br>
{{{1/(1-1/10)=1/(9/10)=10/9}}}<br>
Note the second series is just half of the first, so the sum of the second series is 5/9.<br>
The sum of the original series is then 10/9 + 5/9 = 15/9 = 5/3.<br>
ANSWER: 5/3<br>
Alternatively, we could group the terms in pairs to obtain a single purely geometric series.<br>
{{{1+1/2=3/2}}}
{{{1/10+1/20=3/20}}}
{{{1/100+1/200+3/200}}}<br>
The given series is then equivalent to the series<br>
{{{3/2+3/20+3/200}}}+...<br>
The sum of that series is<br>
{{{(3/2)/(1-1/10)=(3/2)/(9/10)=(3/2)(10/9)=30/18=5/3}}}<br>