Knowing f(x) , is there a general formula for f(1)+f(2)+...f(x) ? I'm guessing it's based on integrals .

Answer 1

In a small number of cases the problem can be solved.

However: Take the much simpler class of problems: "Integrate any given function." This cannot be done, I think.
Here is an example of a difficult integral "The unbounded integral of ln(sin(arctan(exp(973x^13))))", but much more difficult ones can be written down.

In comparison, basic differentiation is a doddle. Most mathematicians could differentiate the expression above, but they wouldn't be interested in doing so.

Answer 2

There are some definite series to help do that. Anyway if you know the general term of a series you can easily find the sum of the series by knowing some definite properties.