Question

For which values of \(r>0\), if any, does \(\sum_{n=1}^{\infty} r^{\sqrt{n}}\) converge? (Hint: \(\left.\sum_{n=1}^{\infty} a_{n}=\sum_{k=1}^{\infty} \sum_{n=k^{2}}^{(k+1)^{2}-1} a_{n} .\right)\)

Short answer

The series converges for values of \(r\) where \(0 < r < 1\).

Step-by-step solution

Understanding the Series

The given series is \( \sum_{n=1}^{\infty} r^{\sqrt{n}} \). To determine for which values of \(r>0\) this series converges, we need to understand how each term behaves as \(n\) increases.

Utilize the Hint to Restructure the Series

The hint suggests restructuring the series: \( \sum_{n=1}^{\infty} a_{n} = \sum_{k=1}^{\infty} \sum_{n=k^{2}}^{(k+1)^{2}-1} a_{n} \). By applying this to our series, consider \( a_{n} = r^{\sqrt{n}} \) and express the series as \( \sum_{k=1}^{\infty} \sum_{n=k^{2}}^{(k+1)^{2}-1} r^{\sqrt{n}} \).

Approximation of Inner Sum

Within each interval \([k^2, (k+1)^2-1]\), every term in the inner sum can be approximated by \(r^k\), as \(r^{\sqrt{n}} \approx r^k\) when \(n\) is close to \(k^2\). Each interval contains approximately \(2k+1\) terms.

Evaluating the Series for Convergence

The series becomes \( \sum_{k=1}^{\infty} (2k+1) r^k \). For this series to converge, \(r\) must be less than 1 (bcause the geometric series is divergent otherwise). Thus, evaluate convergence where the series behaves like \( \sum_{k=1}^{\infty} k r^k \), which converges for \(0 < r < 1\).

Conclusion

The original series \( \sum_{n=1}^{\infty} r^{\sqrt{n}} \) converges when \(0 < r < 1\) due to the convergence of the related series \( \sum_{k=1}^{\infty} kr^k \). At \(r=1\), the series diverges because each partial sum becomes infinite.

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as \( r \). In mathematical terms, a geometric series can be represented as \( a, ar, ar^2, ar^3, \ldots \). A geometric series converges (meaning it approaches a finite value) if its common ratio \( r \) satisfies \( |r| < 1 \). This can be expressed with the formula for the sum of an infinite geometric series:

• \( S = \frac{a}{1 - r} \), when \( |r| < 1 \)
• The series diverges if \( |r| \ge 1 \)

Understanding geometric series is essential when studying the convergence of series, as many complex series can be broken down into simpler geometric components. In the context of our problem, knowing whether a series resembles a geometric series with \(|r| < 1\) tells us if the series converges.

Intervals of Convergence

Intervals of convergence refer to the set of values for which a series converges. For a series with terms depending on a variable, like \( \sum_{n=1}^{\infty} x^n \), it is vital to determine the interval for \( x \) where the series converges. Steps to find the interval involve:

• Establishing the convergence criteria (usually involving limits or bounds).
• Using tests such as the ratio test, root test, or comparison test to find convergence limits.

When considering the series \( \sum_{n=1}^{\infty} r^{\sqrt{n}} \), we determine the interval of convergence by looking at the behavior of the series over values of \( r \). The use of series restructuring and approximations helps simplify finding these intervals.In our case, upon simplification, we found the series converges for \( 0 < r < 1 \). This means that within this interval, the series approaches a limit.

Series Approximation

Series approximation involves simplifying a complex series to a more understandable form. This process uses mathematical techniques to express parts of the series in a simpler manner, making it easier to analyze convergence.The process in our problem was guided by rethinking the series through intervals, as suggested by the given hint. When expressed as \( \sum_{k=1}^{\infty} (2k+1) r^k \,\) each interval acts as a small approximate geometric series.

• Each segment contributes terms that are approximately \( r^k \,\) the key term of a simplified geometric series.
• Approximation allows us to use known values of simpler series to evaluate the entire series.

With each interval containing \( 2k+1 \) terms, these segments collectively have properties similar to geometric series. The approximation displayed that the series behaves like a geometric one with terms like \( k r^k \), crucial for determining convergence.