To find the value of A, we need to evaluate the sum of the series 1 + 1/2 + 1/3 + 1/4 + ... + 1/(2^100-1).

This is a harmonic series, which does not have a simple closed-form expression. However, we can approximate its value using the natural logarithm.

Let's define S = 1 + 1/2 + 1/3 + 1/4 + ... + 1/(2^100-1).

Taking the natural logarithm of both sides, we have:
ln(S) = ln(1) + ln(1/2) + ln(1/3) + ln(1/4) + ... + ln(1/(2^100-1))

Using the property ln(a/b) = ln(a) - ln(b), we can simplify this expression:
ln(S) = ln(1) - ln(2) + ln(1) - ln(3) + ln(1) - ln(4) + ... + ln(1) - ln(2^100-1)

Since ln(1) = 0, all the terms ln(1) can be eliminated:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Using the property ln(a) + ln(b) = ln(a * b), we can further simplify this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))
= ln((2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^(-1))^(-1))
= ln((2^100-1) / (2 * 3 * 4 * ... * (2^100-1)))

Now, we can use the property ln(a/b) = ln(a) - ln(b) to simplify further:
ln(S) = ln(2^100-1) - ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)
= -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1) + ln(2^100-1)

The last term ln(2^100-1) cancels out with the term on the right side, leaving us with:
ln(S) = -ln(2) - ln(3) - ln(4) - ... - ln(2^100-1)

Now, we can use the fact that ln(a) + ln(b) = ln(a * b) to rewrite this expression:
ln(S) = ln(2^(-1) * 3^(-1) * 4^(-1) * ... * (2^100-1)^