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?
I suspect you're having an indexing problem:
If we start at $n=0$, we're looking at the sum $$\left({1\over 5}\right)^0+\left({1\over5}\right)^1+\left({1\over 5}\right)^2+....$$ Here $a_0=1$, so the limit is $${a_0\over 1-r}={1\over {4\over 5}}={5\over 4}.$$
If we start at $n=1$, we're looking at the smaller by $1$ sum $$\left({1\over 5}\right)^1+\left({1\over5}\right)^2+\left({1\over 5}\right)^3+...$$ Here, $a_0={1\over 5}$, not $1$, so the limit is $${a_0\over 1-r}={{1\over 5}\over{4\over 5}}={1\over 4}.$$