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When working with a power series, one common task is to find its derivative. The derivative of a power series can be determined by differentiating each term individually. For the given power series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}\left(x+\frac{1}{2}\right)^{n} \), the derivative is computed as follows: For each term \( \frac{2^{n}}{n^2} \left(x+\frac{1}{2}\right)^{n} \), apply the power rule. This gives \( n \frac{2^{n}}{n^{2}} \left(x+\frac{1}{2}\right)^{n-1} \). The resulting derivative expression becomes \( f^{\prime}(x) = \sum_{n=1}^{\infty} \frac{2^{n}}{n} \left(x+\frac{1}{2}\right)^{n-1} \). By simplifying, we just factor out \( n \) from the denominator, which cancels one \( n \) in the numerator. This step-by-step approach helps manage each component accurately, ultimately providing the derivative of the original power series.

Knowing where a power series converges is critical. The interval of convergence tells us for which values of \( x \) the series results in a finite number. For the series \( \sum_{n=1}^{\infty} \frac{2^n}{n^2} \left(x+\frac{1}{2}\right)^n \), we use criteria like the Ratio Test to deduce this interval for both the original and its derivative.To find this interval, analyze \( \left|\frac{2^{n+1}}{(n+1)^2}\left(x+\frac{1}{2}\right)^{n+1} \div \frac{2^n}{n^2}\left(x+\frac{1}{2}\right)^n\right| \). Simplifying through division yields \( \left|2\left(x+\frac{1}{2}\right)\right| \). This implies convergence when \( \left|x+\frac{1}{2}\right| < \frac{1}{2} \). Thus, the interval is \( -1 < x < 0 \). Understanding this is crucial as it defines the domain where the power series behaves consistently.

The partial sum of a series is a truncated version, which considers only a limited number of terms. In transforming from infinite to partial, this allows for approximation and practical computation of the series.For our derivative function \( f'(x) \), its partial sum up to \( n=4 \) is given by \( \sum_{n=1}^{4} \frac{2^{n}}{n} \left(x+\frac{1}{2}\right)^{n-1} \). This is calculated as:

• \( 2(x+\frac{1}{2})^{0} = 2 \)
• \( 2(x+\frac{1}{2}) \)
• \( \frac{8}{3}(x+\frac{1}{2})^2 \)
• \( 4(x+\frac{1}{2})^3 \)

Adding these terms results in the partial sum: \( 2 + 2(x+\frac{1}{2}) + \frac{8}{3}(x+\frac{1}{2})^2 + 4(x+\frac{1}{2})^3 \). Partial sums are handy in approximating functions when full computation is unfeasible.

The ratio test is a powerful tool for determining the convergence of a series. It is particularly useful for series featuring factorials or exponential terms. The test examines the limit of the ratio of consecutive terms.For the series \( \sum_{n=1}^{\infty} \frac{2^n}{n^2} \left(x+\frac{1}{2}\right)^n \), the ratio test entails:

• Compute \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) where \( a_n = \frac{2^n}{n^2} \left(x+\frac{1}{2}\right)^n \)
• Resulting in \( \lim_{n \to \infty} \left| \frac{2(x+\frac{1}{2})}{1} \right| \)

This simplifies to \( 2|x+\frac{1}{2}| \), which dictates convergence for \( 2|x+\frac{1}{2}| < 1 \). Therefore, \( |x+\frac{1}{2}| < \frac{1}{2} \) results in \( -1 < x < 0 \). This method helps verify convergence simply and effectively, guiding the identification of viable \( x \) values within the series.