Question

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}}$$

Short answer

The given series converges for values of x in the interval [0, 1).

Step-by-step solution

Identify the series and Ratio Test formula.

The given series is: $$\sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}}$$ We need to find the values of \(x \geq 0\) for which the series converges using the Ratio Test. The formula for the Ratio Test is: $$\lim_{k \to \infty} \left |\frac{a_{k+1}}{a_k} \right |$$

Apply the Ratio Test formula on the given series.

We need to find the ratio of consecutive terms (\(\frac{a_{k+1}}{a_k}\)) in the series: $$\frac{a_{k+1}}{a_k} = \frac{\frac{x^{k+1}}{(k+1)^2}}{\frac{x^k}{k^2}}$$ Simplify the expression: $$\frac{a_{k+1}}{a_k} = \frac{x^{k+1}k^2}{x^{k}(k+1)^2}$$

Find the limit of the absolute value of the ratio.

Now, we need to find the limit of the absolute value of the ratio as \(k \to \infty\): $$\lim_{k \to \infty} \left |\frac{x^{k+1}k^2}{x^{k}(k+1)^2} \right |$$ Simplify the expression: $$\lim_{k \to \infty} \left |\frac{x k^2}{(k+1)^2} \right |$$

Evaluate the limit to determine convergence conditions.

Since \(x\) is independent of \(k\), we can take it out of the limit: $$x \lim_{k \to \infty} \left |\frac{k^2}{(k+1)^2} \right |$$ As \(k \to \infty\), the limit converges to 1: $$x \cdot 1 = x$$ For the series to converge, the limit has to be less than 1: $$x < 1$$ Since we are given \(x \geq 0\), the interval of convergence is: $$0 \leq x < 1$$

Conclusion

The series converges for \(0 \leq x < 1\).

Key concepts

Power Series

A power series is an infinite series in the form of \( \sum_{k=0}^{\infty} a_k(x - c)^k \), where \( a_k \) are coefficients and \( x \) is a variable. The power series is centered around \( c \), which means if \( c = 0 \), the series is centered at zero.
In the given problem, we observe that the power series is \( \sum_{k=1}^{\infty} \frac{x^k}{k^2} \). This series doesn't follow the exact form of a traditional power series since the denominator includes \( k^2 \). However, it still demonstrates the structure because \( x^k \) indicates varying powers of \( x \).

• The role of \( x \) here is crucial as it dictates the series' behavior and convergence.
• These types of series frequently appear in calculus when representing functions as series, especially through Taylor and Maclaurin series.

Understanding power series is essential because they allow for the approximation and analysis of complex functions using simpler polynomial forms.

Convergence

Convergence of a series indicates that as you add more terms in the sequence, the sum approaches a definite value. For series like the one we're examining, convergence is dependent on the value of \( x \).

• The Ratio Test is one of the standard methods used to determine convergence, specifically for series with non-negative terms.
• If the limit of the ratio of successive terms \( \left |\frac{a_{k+1}}{a_k} \right | \) is less than 1 as \( k \to \infty \), the series converges.

In our specific problem, by applying the Ratio Test, we found that the series converges for \( 0 \leq x < 1 \). This means if you substitute any value of \( x \) within these bounds, adding up the infinite number of terms will sum up to a specific number. Outside this interval, the series diverges, meaning the sum doesn't settle at a single value.

Limit Evaluation

Limit evaluation is crucial for using the Ratio Test in determining the convergence of a series. It involves simplifying and calculating the limit of the terms' ratio as \( k \to \infty \).
Applying the Ratio Test to the series \( \sum_{k=1}^{\infty} \frac{x^k}{k^2} \) required finding the limit of \( \frac{a_{k+1}}{a_k} \). Here's what that process looked like:

• First, find the expression for consecutive terms, which simplifies the ratio.
• Next, isolate terms involving \( x \) since they don't depend on \( k \), making computation of the limit with respect to \( k \) simpler.
• Finally, calculate the limit, where in our case, the expression simplified such that the limit was found to be \( x \).

Understanding and evaluating limits is essential not only for the Ratio Test but also in deeper aspects of calculus, such as solving differential and integral equations. Proper limit manipulation ensures accurate results and conclusions regarding convergence or divergence.