Why don't infinite sums make any sense? Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4.  However, how is this possible, considering that each additional sum is smaller than the previous sum?  If you begin to do the math by hand, calculating and adding each sum individually, you get something like .249984...etc... where it appears that the number will extend infinitely to the right, adding decimal places and occasionally revising numbers later to the right of the decimal, but the additional sums should never be large enough to affect the initial answers in the decimal like the .2 part of the answer?  It seems very counterintuitive.
There seems to be a basic contradiction between the expected answer and the answer of the equation by a large margin of difference. Can anyone explain this?

If you try working in base 5 rather than base 10 then it will become more obvious.
It's kind of like how using 1/9 instead of 1/5 gives you 0.99999... = 1 which I hope you agree with.
