I have two issues related to finite sum of infinite series,
1) How you would to describe 2 when you talk about the infinite geometric series 1+ 1/2 + 1/4 + 1/8 + .....
2) How you would compare using partial sums in following cases,
1-1+1-1+...... = 1 -(1-1+1-1+...) which gives 1-1+1-1+... = 1/2 With the method we apply to express recurring decimals as fractions ( ratio between two integers)
- Which way is correct , 2 is the sum of infinite number of terms or 2 is the limit of sum of n number of terms as n tends to infinity. In exam papers I have seen students are asked to find sum of infinite number of terms.
2)As we know 1/2 can not be accepted as the sum of 1-1+1-.... then what about applying the same method in recurring decimals, there we take x as the recurring part and subtract two equations to remove x to get rational form of the number. Where x is sum of infinite series.