Question

As in Example 1, use the ratio test to find the radius of convergence \(R\) for the given power series. $$ \sum_{n=1}^{\infty} \frac{t^{n}}{n^{2}} $$

Short answer

Question: Determine the radius of convergence, \(R\), for the given power series: $$ \sum_{n=1}^{\infty} \frac{t^{n}}{n^{2}} $$ Answer: The radius of convergence, \(R\), for the given power series is 1.

Step-by-step solution

Identify the terms \(a_n\) and \(a_{n+1}\)

The given series is: $$ \sum_{n=1}^{\infty} \frac{t^{n}}{n^{2}} $$ Thus, we have \(a_n = \frac{t^n}{n^2}\) and \(a_{n+1} = \frac{t^{n+1}}{(n+1)^2}\).

Calculate the limit using the ratio test

The limit we need to calculate is: $$ \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left|\frac{\frac{t^{n+1}}{(n+1)^2}}{\frac{t^n}{n^2}}\right| $$ Now, simplify the fraction within the absolute value: $$ \lim_{n \to \infty} \left|\frac{t^{n+1}n^2}{(n+1)^2t^n}\right| $$ Note that \(t^{n+1} = t^n \cdot t\), so the expression can be further simplified: $$ \lim_{n \to \infty} \left|\frac{t^n \cdot t \cdot n^2}{(n+1)^2t^n}\right| = \lim_{n \to \infty} \left|\frac{t \cdot n^2}{(n+1)^2}\right| $$

Result of the limit

The limit can be solved by observing what happens as \(n\) goes to infinity: $$ \lim_{n \to \infty} \left|\frac{t \cdot n^2}{(n+1)^2}\right| $$ As \(n\) approaches infinity, the term \(n^2\) dominates over the term \((n+1)^2\), we can then simplify it as: $$ \lim_{n \to \infty} |t| $$

Determine the radius of convergence

For the given power series to converge using the ratio test, we need the result of the limit to be smaller than 1: $$ |t| < 1 $$ This inequality indicates that the radius of convergence \(R\) is equal to 1.

Key concepts

Ratio Test

Understanding the Ratio Test is crucial for determining the convergence of a series, particularly power series. It involves comparing the absolute value of the ratio of successive terms in a series. If this ratio approaches a limit less than one as the series progresses to infinity, the series converges. Let's consider the essential steps in applying the Ratio Test:

• Identify the general term of the series, commonly denoted as (this is the elements that are being summed up).
• Calculate the limit of the absolute value of the ratio of consecutive terms, namely .
• If the limit is less than one, the series converges; if it is greater than one, it diverges. A limit equal to one does not provide information on convergence or divergence and occasionally requires applying other tests.

Using the Ratio Test allows us to find the radius of convergence for power series, which tells us the interval in which the series converges around the center point of the series.

Power Series Convergence

The convergence of a Power Series is a fundamental concept in calculus. A power series looks like this: , where is a variable and are constants. Here's what you need to know about power series convergence:

• The series can converge absolutely, meaning it converges for all within the radius of convergence .
• If the radius of convergence is , then the power series converges for all in real numbers.
• On the boundary of the convergence interval, additional work is needed to determine convergence.

Understanding where a power series converges is vital for further applications, such as solving differential equations or evaluating complex integrals.

Limit Calculation

The process of Limit Calculation is one of the cornerstones of calculus. When dealing with sequences or functions, limits help us understand behavior as we approach a specific point. Here are some tips on performing limit calculations:

• Factorization can simplify expressions before taking limits.
• L'Hôpital's Rule can be applied when you encounter a form such as .
• Understanding the behavior of polynomials, exponentials, and other functions at infinity is essential.
• Remember that the limit of a sequence as only depends on terms that get large as .

In our exercise, the limit calculation was significantly simplified by recognizing that as , the expression behaves like itself, leading us to a clear answer for the convergence test.

Absolute Value Simplification

Working with Absolute Value Simplification can sometimes be tricky, but it's a fundamental skill for solving many mathematical problems. Here are some points to keep in mind:

• The absolute value of a real number is its non-negative value, regardless of its original sign.
• In limit calculations, absolute values can often be simplified if we know the sign of the variables involved.
• Be cautious of the fact that removing the absolute value incorrectly can lead to false results.

In our exercise example, the absolute value simplification was straightforward because the limit calculation showed the variable was non-negative. Accordingly, the absolute value simplification did not affect the convergence result.