Finite sum of infinite series

Question

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)

1. 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.

Answer 1

"Infinite sum" is in common use, so it should be acceptable to say that $2$ is the value of the infinite sum $1+\frac{1}{2}+\frac{1}{4}+\ldots$. But students need to be very clear that the value of an infinite sum is defined to be the limit of the partial sums, assuming that limit exists, and that if the limit doesn't exist, the infinite sum is undefined. They should also be well aware that operations that would not change the value of a finite sum, such as rearranging terms, can change the value of an infinite sum unless it is absolutely convergent. They should be aware of the Riemann rearrangement theorem, which says that the terms of a conditionally convergent series can be arranged to give any value whatsoever, including $\infty$ and $-\infty$.

You ask to compare the calculation $$ x=1-1+1-1+1-\ldots=1-(1-1+1-1+\ldots)=1-x, $$ which leads to $x=1-x$ or $x=\frac{1}{2}$, and calculations like $$ x=0.3333\ldots=-3+3.3333\ldots=-3+10x, $$ which leads to $x=-3+10x$ or $x=\frac{1}{3}$.

The issue here is that the first sum is divergent since the partial sums do not have a limit, making the entire calculation illegitimate (What is $x$ here?) In contrast, the second sum is convergent. It is only because of the convergence that we can meaningfully write $x=0.3333\ldots$. Without convergence $x$ would be undefined. Once we have convergence we can prove the other needed statement, namely that $3.3333\ldots=10x$. All infinite decimal numbers can be seen to be convergent by, for example, using the comparison test with the convergent geometric series $0.999\ldots$.