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The given expression \((1 + 2x)^{-\frac{1}{2}}\) suggests that you are looking to find the sum to infinity of a series derived from this expression. This is typically done by expanding the expression into a binomial series.

### Step-by-Step Solution:

We can express \((1 + 2x)^{-\frac{1}{2}}\) as a binomial series, but this expansion is valid only when \(|2x| < 1\), which implies \(|x| < \frac{1}{2}\). The binomial expansion for \((1 + u)^n\), where \(n = -\frac{1}{2}\) and \(u = 2x\), is:

\[
(1 + 2x)^{-\frac{1}{2}} = 1 + \binom{-\frac{1}{2}}{1}(2x) + \binom{-\frac{1}{2}}{2}(2x)^2 + \binom{-\frac{1}{2}}{3}(2x)^3 + \dots
\]

The general binomial coefficient for negative powers is:

\[
\binom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!}
\]

For \(n = -\frac{1}{2}\), the first few terms of the series expansion will look like:

\[
(1 + 2x)^{-\frac{1}{2}} = 1 - x + \frac{3}{2}x^2 - \frac{5}{2}x^3 + \dots
\]

### Sum to Infinity

The infinite series derived from this expansion is:

\[
1 - x + \frac{3}{2}x^2 - \frac{5}{2}x^3 + \dots
\]

This is the binomial series expansion of \((1 + 2x)^{-\frac{1}{2}}\). The sum of this series continues indefinitely, but the series is convergent only for \(|x| < \frac{1}{2}\).

Would you like further details or clarifications? Here are some related questions:

1. How does the binomial theorem apply to negative or fractional powers?
2. What are the conditions for the convergence of a binomial series?
3. How would you expand the expression for a different value of \(n\)?
4. Can you apply this approach to other functions, like \((1 - x)^{-\frac{1}{2}}\)?
5. How do you compute the individual binomial coefficients for negative powers?

**Tip:** When dealing with binomial expansions for fractional exponents, always check the conditions for convergence, typically related to the absolute value of the variable being less than 1 for the series to converge.

Find the sum to infinity of the series (1 + 2x)^(-1/2).

Learn how to find the sum to infinity of the series expansion for (1 + 2x)^(-1/2). This problem covers binomial expansion for fractional powers, convergence conditions, and algebraic techniques, suitable for Grades 11-12.

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