Question

How do you write the series `-2+ 1+ 6+ 13+ 22+...` using summation notation?

Answer 1

`sum_(n=1)^oo (n^2-3)`

Explanation:

No finite sequence of terms determines an infinite series, unless you have more information about the sequence - e.g. arithmetic, geometric, quadratic, cubic, etc.

However, note that the differences between consecutive terms of the given sequence are consecutive odd numbers:

`3, 5, 7, 9`

...at least as far as it goes.

If this pattern continues, then note that there's another sequence with these differences:

`1, 4, 9, 16, 25`

which is recognisable as:

`1^2, 2^2, 3^2, 4^2, 5^2`

Subtracting `3` from each term, we arrive at the given sequence.

So a formula that matches the given sequence is:

`a_n = n^2-3`

We can write the sum of the sequence, i.e. the series, as:

`sum_(n=1)^oo a_n = sum_(n=1)^oo (n^2-3)`

This does not converge.