Question

How do you use a calculator to find the sum of `Sigma (-1)^k/(k+1)` where k is [0,4]?

Answer 1

Calculate the expression at 0, 1, 2, 3, 4 and add them up!
`sum_(0,4) ((-1)^k)/(k + 1) = 0.783`

Explanation:

`sum_(0,4) ((-1)^k)/(k + 1)` = `((-1)^0)/(0 + 1) + ((-1)^1)/(1 + 1) + ((-1)^2)/(2 + 1) + ((-1)^3)/(3 + 1) + ((-1)^4)/(4 + 1)`

`sum_(0,4) ((-1)^k)/(k + 1)` = `1 -1/2 + 1/3 - 1/4 + 1/5`

You may see a pattern developing for the expression in general. For this range, the final value is:

`sum_(0,4) ((-1)^k)/(k + 1) = 0.783`

The "SUM" means addition of all of the factors. The final value depends on the resolution of the scale, which not defined explicitly here. It cannot be infinitely small, or the sum would also become infinite.
w.differencebetween.com/difference-between-integration-and-vs-summation/

Other examples here:
https://socratic.org/calculus/introduction-to-integration-/sigma-notation-1