Evaluate 0.2175 using the sum of infinite geometric series?

Σ = a / (1 - r)

As presented, your question is MEANINGLESS !!

When |r| < 1, sum of infinite geometric series = a₁ / (1 - r)

====
Case I: S = 0.2175217521752175……
Then, S = 0.2175 + 0.2175×(0.0001) + 0.2175×(0.0001)² + ……
S = 0.2175 / (1 - 0.0001) = 0.2175/0.9999 = 2175/9999

====
Case II: S = 0.2175175175175175……
Then, S = 0.2 + [0.0175 + 0.0175×(0.001) + 0.0175×(0.001)² + ……]
S = 0.2 + [0.0175 / ( 1 - 0.001)]
S = 0.2 + [0.0175 / 0.999]
S = 0.2 × (9990/9990) + (175/9990)
S = 2173/9990

====
Case III: S = 0.2175757575……
Then, S = 0.21 + [0.0075 + 0.0075×(0.01) + 0.075× (0.01)² + …...]
S = 0.21 + [0.0075 / (1 - 0.01)]
S = 0.21 × (9900/9900) + (75/9900)
S = 2154/9900
S = 359/1650

====
Case IV: S = 0.21755555……
Then, S = 0.217 + [0.0005 + 0.0005×(0.1) + 0.0005×(0.1)² + ……]
S = 0.217 + [0.0005 / (1 - 0.1)]
S = 0.217 × (9000/9000) + (5/9000)
S = 1958/9000
S = 979/4500

====
Case V: S = 0.2175 (exactly)
Then, S = 2175/10000
S = 87/400

0.2175 = 2/10 + 1/100 + 7/1000 + 5/10000

0.2175=0.21+0.0075+0.000075+.......
=0.21+75\10⁴(1+1/10²+1/10⁴+......)
=0.21+(75/10⁴)×1/1-1/10²)
0.21+(71/10⁴)×10²/99
=21/100+(3/4)×(1/99)
=21/100+1/132
=359/1650

Why are you answering your own question ? Playing games, are we ? 0.1275 IS an evaluation, there is nothing more to evaluate.

You have no need to have recourse to any series, geometrical or anything else, to convert it to a rational fraction, certainly not any infinite one. Infinite bullshit, definitely.

　
We evaluate expressions or sums, not numbers.

Numbers are what we use to evaluate. They are the end result. We don't evaluate numbers because they already have a value.