Final answer:
To find the sum of the given series 1+ 1/{2¹/² }+ 1/2+ 1/{2³/²}+ ..., we can use the formula for the sum of an infinite geometric series.
Step-by-step explanation:
To find the sum of the given series 1+ 1/{2¹/² }+ 1/2+ 1/{2³/²}+ ..., we can rewrite each term with a common denominator and then simplify.
First, let's express each term with a common denominator of 2.
The series can be rewritten as 2/2 + 1/{2¹/² } + 1/2 + 1/{2³/²} + ..., where 2/2 is equal to 1.
Now, let's simplify each term.
The series becomes 1 + 1/{√2} + 1/2 + 1/{√8} + ...
We can observe that each term can be represented as 1/{√(2^n)}, where n starts from 0.
To find the sum of the series, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r), where a is the first term and r is the common ratio.
In this case, a = 1 and r = 1/√2.
Substituting the values into the formula, we get:
Sum = 1 / (1 - 1/√2).
Simplifying further, we obtain:
Sum = 1 / (1 - 1/√2) = (√2) / (√2 - 1).