I'm not sure if you are asking for an answer or a rhetorical question? I'll assume the former.
Your terms are bit jumbled, so let's keep it simple: you're asking how to prove if an infinite sum converges and what its value is. Convergence proofs require analytic thought: meaning there may not be an immediate look-up. You need to convert the problem into the known corpus of convergent sums or use one of many tests (bounds test, integral test, etc) to show it converges analytically. Which you only learn through experience and memorization (unless you want to re-prove hundreds of series... maybe you do!) Fortunately this one is easily re-written as a known convergent sum.
First, you missed a term in your sum (9), re-written here:
sum(n=1..inf) 9 * 10^-n
Step 1: you pull out the 9 and it becomes 1/10+1/100+1/1000...
Step 2: Then we shift to n=0 by subtracting 1/10^0 from the series so that it is in the form n=0..k-1
Step 3: Now we've got ourselves a geometric series of just 1/10^n .. wikipedia does a great job explaining the sum convergence for GS from n=0...inf: https://en.wikipedia.org/wiki/Geometric_series
Your terms are bit jumbled, so let's keep it simple: you're asking how to prove if an infinite sum converges and what its value is. Convergence proofs require analytic thought: meaning there may not be an immediate look-up. You need to convert the problem into the known corpus of convergent sums or use one of many tests (bounds test, integral test, etc) to show it converges analytically. Which you only learn through experience and memorization (unless you want to re-prove hundreds of series... maybe you do!) Fortunately this one is easily re-written as a known convergent sum.
First, you missed a term in your sum (9), re-written here:
sum(n=1..inf) 9 * 10^-n
Step 1: you pull out the 9 and it becomes 1/10+1/100+1/1000...
Step 2: Then we shift to n=0 by subtracting 1/10^0 from the series so that it is in the form n=0..k-1
1/10^0 + 1/10 + 1/100 + 1/1000 + ... + 1/10^-n - 1/10^0
Step 3: Now we've got ourselves a geometric series of just 1/10^n .. wikipedia does a great job explaining the sum convergence for GS from n=0...inf: https://en.wikipedia.org/wiki/Geometric_series
Step 4: compute geometric convergence
(1-r^n)/(1-r) = (1-(1/10)^n)/(1-1/10) = 1/(1-1/10) = 10/9
So we have 10/9 as the solution to Sum[n=0...inf](1/10^n)
Step 5: the remaining arithmetic
Now subtract our 1/10^0 ... and then * 9 = 1