Question

Suppose that \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=p .\) For which values of \(r>0\) is \(\sum_{n=1}^{\infty} r^{n} a_{n}\) guaranteed to converge?

Short answer

The series converges for values of \( r > 0 \) such that \( r < \frac{1}{p} \).

Step-by-step solution

Understand the ratio test

The ratio test for series convergence provides a way to determine if a series \( \sum a_n \) converges. For a series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = p \), the series converges if \( p < 1 \).

Redefine the series

We need to find for which \( r > 0 \) the series \( \sum_{n=1}^{\infty} r^n a_n \) converges. This can be rewritten as \( \sum_{n=1}^{\infty} b_n \), where \( b_n = r^n a_n \).

Apply the ratio test to the new series

Apply the ratio test to the series \( \sum b_n = \sum r^n a_n \). Evaluate \( \lim_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| = \lim_{n \to \infty} \left| \frac{r^{n+1} a_{n+1}}{r^n a_n} \right| = |r| \cdot \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |r|p \).

Determine convergence condition

For the series \( \sum b_n \) to converge, \( \lim_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| = |r|p < 1 \). Thus, the series converges when \( |r|p < 1 \), or equivalently, \( r < \frac{1}{p} \).

Conclusion for positive r

Since \( r > 0 \) and the inequality \( r < \frac{1}{p} \) must hold, \( r \) must be less than \( \frac{1}{p} \) for the series \( \sum_{n=1}^{\infty} r^n a_n \) to converge.

Key concepts

Ratio Test

The ratio test is a handy tool for determining the convergence of an infinite series. Imagine you have a series like \( \sum a_n \). To use the ratio test, we compute the ratio of successive terms: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). This ratio tells us how the terms of the series behave as \( n \) becomes very large.
If this limit \( p \) is:

• Less than 1, the series converges absolutely.
• Greater than 1, the series diverges.
• Equal to 1, the test is inconclusive.

By applying the ratio test to \( \sum a_n \), we found \( p \). For a new series, say \( \sum b_n = \sum r^n a_n \), the ratio test helps us determine the conditions under which this series converges by evaluating the limit \( |r|p < 1 \).
Feel free to apply this technique whenever you're faced with an infinite series with complex terms and need a straightforward method to check convergence.

Infinite Series

An infinite series is essentially the sum of infinitely many terms. Think of it as the sum \( a_1 + a_2 + a_3 + \ldots \). Understanding whether or not this kind of sum converges (adds up to a finite value) is fundamental in mathematics.
When considering convergence:

• We investigate if the sum of the terms approaches a specific, finite number as we add more terms.
• If it does converge, it means, practically, we can predict the total value by adding enough terms.
• If it diverges, adding terms keeps growing the total without any bound, never settling on one value.

In exercises like ours, often infinite series are modified, like when we consider \( \sum r^n a_n \). The concept of convergence decides if this new series has a meaningful, finite sum, and tools like the ratio test can be applied to investigate further.
Understanding infinite series helps in numerous fields of math, showing up in calculus, solving equations, and even predicting outcomes in real-world scenarios.

Limit of a Sequence

At the heart of understanding series and their convergence is the concept of the limit of a sequence. A sequence is an ordered list of numbers, and finding its limit involves seeing what value the sequence approaches as we move further along the list.
For a sequence \( a_n \):

• If \( \lim_{n \to \infty} a_n = L \), it means the terms get closer to \( L \) as \( n \) becomes very large.
• With no limit, the sequence doesn't settle down, instead it may keep growing or oscillating.

In the context of our exercise, understanding the limit of \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = p \) provides the crucial gateway to applying the ratio test. We used the limit to determine if the series \( \sum a_n \) converges.
Mastering the concept of limits is critical as it forms the backbone of calculus and other advanced math topics, focusing on behavior as things approach infinity.