1-1/9+1/25-1/49+1/81 in sigma notation?
Notice that each of our terms seems to be divided by a perfect square. That is,
We also note that we initially start with a 1, then we subtract our next term. We add the term after that and subtract the term following that. That is, we alternate adding and subtracting terms.
Using these two facts, we wish to devise a formula
We will first figure out the denominator. We create a small table in which we have our value of
1 | 1
We can see now that our denominator can be represented as
We simply need to build in the alternating addition and subtraction. We do this by adding the term
Thus, our answer is