Sum of a series of a number raised to incrementing powers

Two great math courses that deal with sums and sequences such as the one you have defined are Real Analysis and basic Single-Variable Calculus. This problem specifically deals with geometric progression. Yes, you do learn some in high school, but not that much. Real Analysis is a subject that gives you a more structured intuition for these types of problems.

The solution to your problem is this by a geometric sum:

$$2^0+2^1+2^2+2^3+\cdot\cdot\cdot+2^n=\frac{2^{n+1}-1}{2-1}=\boxed{2^{n+1}-1}.$$

That is a geometric sum:

$${ 2 }^{ 0 }+{ 2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 3 }+...+{ 2 }^{ n }=\frac{2^{n+1}-1}{2-1}$$

It is taught at highschools.

There is no need to estimate, the exact answer is $2^{n+1}-1$.

This is basic algebra as taught in schools.