To find the value of A, we need to evaluate the sum of the given series.

The general term of the series is 1/(1+2+3+...+n), where n ranges from 1 to 2021.

We can rewrite the denominator as the sum of an arithmetic series:
1+2+3+...+n = n(n+1)/2

So, the general term becomes 1/(n(n+1)/2).

Now, let's evaluate the sum of the series:

A = 1/(1(1+1)/2) + 1/(2(2+1)/2) + 1/(3(3+1)/2) + ... + 1/(2021(2021+1)/2)

Simplifying the denominators, we get:

A = 2/2 + 2/6 + 2/12 + ... + 2/(2021*2022/2)

A = 1 + 1/3 + 1/6 + ... + 1/(2021*2022/2)

Now, let's find a common denominator for all the terms:

A = (1*2022)/(1*2022) + (1*2022)/(3*2022) + (1*2022)/(6*2022) + ... + (1*2022)/(2021*2022/2)

A = (2022 + 2022/3 + 2022/6 + ... + 2022/(2021/2))

A = 2022(1 + 1/3 + 1/6 + ... + 1/(2021/2))

Now, let's simplify the sum of the reciprocals:

1 + 1/3 + 1/6 + ... + 1/(2021/2) = 1 + 1/2 + 1/3 + ... + 1/2021

This is a well-known series called the Harmonic series, and its sum is approximately equal to the natural logarithm of the number of terms:

1 + 1/2 + 1/3 + ... + 1/2021 ≈ ln(2021)

Therefore, the value of A is approximately:

A ≈ 2022 * ln(2021)