1 - 1 + 1 - 1 + 1 - 1 + 1,.... till infinity, what's the result?

Question

is it a violation of associative rule?

(a + b) + c = a + (b + c)

well you see:

(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + .... = 0

while

1 + ( - 1 + 1) + ( - 1 + 1) + ( - 1 + 1) +.... = 1

Just curious really...

****The contradiction you point out is one reason why it is not possible to give a reasonable meaning to every infinite sum. Infinite sums are essentially meaningless until a careful definition is given, and the usual way of defining an infinite sum as a kind of limit leaves this one out in the cold as simply being nonsense.
#### How about the geometric series:
1+1/2+1/4+1/8+...... = 2
Infinite sums have solution if the limit of sum is convergent

Answer 1

Your assumption is wrong. The key words are "till infinity". The answer is =0

Answer 2

The contradiction you point out is one reason why it is not possible to give a reasonable meaning to every infinite sum. Infinite sums are essentially meaningless until a careful definition is given, and the usual way of defining an infinite sum as a kind of limit leaves this one out in the cold as simply being nonsense.

Answer 3

it shud be obvious that the series comes to 0 for even number of terms and 1 for odd number of terms.

Since, infinity cannot be said to be even or odd, this is an indeterminable series. The infinite sum of any series can be said to exist only if it converges .. i.e., as number of terms increase, the term tends to become 0 and the sum gets closer and closer to some value ... in this case, that value is the sum.

In this series, the terms do not tend to become 0 and hence, the series does not have a sum for infinite terms .. so the answer is indeterminable.

As for the associative rule, wat u get is 0+0+0+ ... which is nothing but 0*infinity and u hv assumed it to be 0 which is wrong .. as this can take any value and not necessarily 0. So, concluding it to be 0 is wrong.

Answer 4

Both answers are correct however you must remember that you are considering a different number of terms in each expression - the first one has an even number - the second an odd number - adding the next (+-1) to either of the sequences will produce the same answer as the other one to the same number of terms no matter which method of associating you use.

Answer 5

This series is called Grandi's series. It is a divergent series as it does not approach any number as the sequence of partial sums is 1,0,1,0,1,0....
Divergent series such as this can be assigned a sum using a technique called Cesàro summation. The Cesàro sum of this series is 1/2.

Answer 6

The associative rule has NO place with infinite series.It is not valid.
The only conclusion is that this is an oscilating series as the partial sums are 1,0,1,0 and so on

Answer 7

answer can be plus minus 1or 0