Solution:

Given, the series is: sqrt(2) 1 / (sqrt(2)) 1 / (2sqrt(2)) ...

Let Sn be the sum of the first n terms of the series.

Then, S1 = sqrt(2), S2 = sqrt(2) + 1 / sqrt(2), S3 = sqrt(2) + 1 / sqrt(2) + 1 / (2sqrt(2)), ...

To find the sum of the series, we need to take the limit as n approaches infinity of Sn.

Step 1: Finding a pattern

Let's try to find a pattern in the series by writing out the first few terms:

S1 = sqrt(2)

S2 = sqrt(2) + 1 / sqrt(2)

S3 = sqrt(2) + 1 / sqrt(2) + 1 / (2sqrt(2))

S4 = sqrt(2) + 1 / sqrt(2) + 1 / (2sqrt(2)) + 1 / (4sqrt(2))

From this, we can see that the nth term of the series is given by:

1 / (2^(n-1) * sqrt(2))

Step 2: Finding the sum of the series

Using the formula for the nth term, we can write the sum of the series as:

Sn = sqrt(2) + 1 / sqrt(2) + 1 / (2sqrt(2)) + ... + 1 / (2^(n-1) * sqrt(2))

Multiplying the numerator and denominator of each term by sqrt(2), we get:

Sn = (sqrt(2) * sqrt(2) + 1 + 1 / 2 + ... + 1 / (2^(n-1))) / sqrt(2)

Now, the sum of the series in the numerator is a geometric series with first term 1 and common ratio 1/2. Thus, the sum of this series is:

1 / (1 - 1/2) = 2

Substituting this back into the expression for Sn, we get:

Sn = (sqrt(2) * sqrt(2) + 2) / sqrt(2) = sqrt(2) + 2

Step 3: Taking the limit as n approaches infinity

Finally, taking the limit as n approaches infinity, we get:

lim (n->inf) Sn = lim (n->inf) (sqrt(2) + 2) = sqrt(2) + 2

Therefore, the sum of the series is sqrt(2) + 2.

Hence, the correct option is (b) 2.