Question

Find the radius of convergence of $$\sum\left(1+\frac{1}{k}\right)^{k^{2}} x^{k}$$

Short answer

Answer: The radius of convergence for the series is 0.

Step-by-step solution

Apply Ratio Test

To find the radius of convergence of a series, we need to apply the Ratio Test, which requires calculating the limit as \(k\) tends to infinity of: $$\frac{\left|a_{k+1}\right|}{\left|a_k\right|}$$ In our case, the given series is: $$\sum\left(1+\frac{1}{k}\right)^{k^{2}}x^{k}$$ The \(k\)-th term of the series, \(a_k\), is \(\left(1+\frac{1}{k}\right)^{k^{2}}x^{k}\). Then, the term \(a_{k+1}\) is \(\left(1+\frac{1}{k+1}\right)^{(k+1)^{2}}x^{k+1}\).

Compute the Ratio

Now we need to find the ratio of \(a_{k+1}\) and \(a_k\): $$\frac{\left|\left(1+\frac{1}{k+1}\right)^{(k+1)^{2}}x^{k+1}\right|}{\left|\left(1+\frac{1}{k}\right)^{k^{2}}x^{k}\right|}$$ Simplify the expression: $$\frac{\left(1+\frac{1}{k+1}\right)^{(k+1)^{2}}}{\left(1+\frac{1}{k}\right)^{k^{2}}} \cdot \left|\frac{x^{k+1}}{x^k}\right|$$

Further Simplifications

The terms containing \(x\) can be further simplified as follows: $$\frac{x^{k+1}}{x^k} = \frac{x^k\cdot x}{x^k} = x$$ The updated ratio is: $$\frac{\left(1+\frac{1}{k+1}\right)^{(k+1)^{2}}}{\left(1+\frac{1}{k}\right)^{k^{2}}} \cdot |x|$$

Apply the Ratio Test

Applying the Ratio Test, we compute the limit as \(k\) tends to infinity: $$\lim_{k\to \infty} \frac{\left(1+\frac{1}{k+1}\right)^{(k+1)^{2}}}{\left(1+\frac{1}{k}\right)^{k^{2}}} \cdot |x|$$ We know that \(\lim_{k\to \infty} \left(1+\frac{1}{k}\right)^{k} = e\). Therefore, our limit becomes: $$\lim_{k\to \infty} \frac{e^{(k+1)^2}}{e^{k^2}}\cdot |x|$$ Dividing the exponents, we arrive at: $$\lim_{k\to \infty} e^{(k+1)^2 - k^2}\cdot |x|$$ Then, expanding \((k+1)^2\) gives: $$\lim_{k\to \infty} e^{k^2 + 2k + 1 - k^2}\cdot |x|$$ The limit is now: $$\lim_{k\to \infty} e^{2k + 1}\cdot |x|$$

Determine the Radius of Convergence

The series converges if the limit is less than 1: $$e^{2k + 1}\cdot |x| < 1$$ Since \(e^{2k+1}\) approaches infinity as \(k\) tends to infinity, the only way for the inequality to hold is when \(|x|=0\). Thus, the radius of convergence is: $$R = 0$$ The radius of convergence for the series \(\sum\left(1+\frac{1}{k}\right)^{k^{2}}x^{k}\) is equal to 0.

Key concepts

Ratio Test

The Ratio Test is a handy tool for determining the convergence of an infinite series. It involves looking at the ratio of successive terms in the series. For a series \( \sum a_k \), the Ratio Test examines the limit of \( \left| \frac{a_{k+1}}{a_k} \right| \) as \( k \) approaches infinity.

The basic idea is simple:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In the given exercise, we apply this test to the series \( \sum\left(1+\frac{1}{k}\right)^{k^{2}}x^{k} \) to find the radius of convergence.

Power Series

A power series is a series of the form \( \sum a_k x^k \), where \( x \) is a variable and \( a_k \) are coefficients. These are often used in mathematical analysis because of their ability to represent functions as infinite sums.

Power series are powerful because they allow us to approximate functions to any desired accuracy by considering only a finite number of terms. The domain where a power series converges to a well-defined value is governed by its radius of convergence, which is often determined by tests such as the Ratio Test.

In our case, we have a special power series given by \( \sum \left(1+\frac{1}{k}\right)^{k^{2}} x^{k} \), and we are interested in finding how it behaves with respect to convergence.

Convergence Criteria

Convergence criteria help decide whether a given series converges or diverges. When it comes to power series, the term 'radius of convergence' becomes central.

In simpler terms, the radius of convergence specifies the interval within which all values of \( x \) will result in a convergent series. Beyond this interval, the series fails to converge and behaves chaotically.

Through the Ratio Test, which is part of these criteria, one can ascertain the boundaries of convergence. In the example provided, the calculation led us to conclude that the radius of convergence was zero. This means the series only converges at \( x = 0 \).'.

Limit Calculation

Limit calculation is the process of determining the value that a function approaches as the input approaches some point. In the context of the Ratio Test, we focus on limits to find out how the ratio of consecutive terms behaves as the term number increases indefinitely.

For the series \( \sum\left(1+\frac{1}{k}\right)^{k^{2}}x^{k} \), the computation involves examining the complex limits of expressions involving exponential functions, as seen when calculating \( \lim_{k\to \infty} e^{2k + 1} \cdot |x| \).

Here, the calculation showed that \( e^{2k + 1} \) explodes to infinity, so to keep \( e^{2k+1}\cdot |x| < 1 \), the absolute value \( |x| \) must necessarily be zero. This exact determination of limit behavior underpins why the radius of convergence is unique and, in our case, exactly zero.