Evaluate the infinite sum:
With the given condition that:
Please show your work too — thanks in advance!

Consider the k-th partial sum,

More compactly,

(this is just another case of a similar sum you asked about a while ago [24494877])

The infinite sum is the limit of the partial sum as k goes to infinity. We have

since the non-constant terms in the limit converge to 0.

Alternatively, recall that for |x| < 1, we have

Differentiating both sides gives

also valid for |x| < 1. Take x = 1/π and you get the sum you want to compute.

step-by-step explanation:

how many centimeters did the level of the salsa go down by?