How do you find an function equivalent to a summation?

Question

Without intuitively knowing that $\left[1+2+\cdots+n\right]$ is equivalent to $\frac{n^2+n}{2}$, how would I find a function that represents $\sum_{i=1}^{n} i$ ?

If possible, I would like to know a general way to find these solutions.

Answer 1

There is no general answer. For example, no closed form is known for $$\sum_{i=0}^n \binom{n}{i}^3.$$

There is a large body of specific technique, and there is a fascinating theory that solves many such problems that you can read about in the book A=B by Petkovsek et al., which is available online for free.

When the summand is a polynomial, as in your example, there is a solution; you should look into Faulhaber's formula. This was discovered independently by Johann Bernoulli and Seki Kōwa, both around 1700.

The book Concrete Mathematics by Graham et al. is not too hard, and a large portion of it is devoted to techniques for evaluating summations. For example, it presents seven or eight different ways to obtain a closed form for $\sum_{i=1}^n i^2$, because this example is not too hard and illustrates the various techniques.