Question

The kth term of each of the following series has a factor \(x^{k}\). Find the range of \(x\) for which the ratio test implies that the series converges. $$ \sum_{k=1}^{\infty} \frac{x^{k}}{k !} $$

Short answer

The series converges for all real values of \( x \).

Step-by-step solution

Identify the general term

The general term of the series is given as \( a_k = \frac{x^k}{k!} \).

Apply the Ratio Test

To apply the ratio test, find \( \frac{a_{k+1}}{a_k} \):\[\frac{a_{k+1}}{a_k} = \frac{\frac{x^{k+1}}{(k+1)!}}{\frac{x^k}{k!}} = \frac{x^{k+1} \cdot k!}{x^k \cdot (k+1)!} = \frac{x}{k+1}\]

Evaluate the Limit of the Ratio

Find the limit as \( k \to \infty \):\[\lim_{k \to \infty} \left| \frac{x}{k+1} \right| = \lim_{k \to \infty} \frac{|x|}{k+1} = 0\]

Conclusion of the Ratio Test

Since the limit is 0, which is less than 1 for any \( x \), the series converges for all real numbers \( x \). The ratio test implies convergence when \( | \text{limit} | < 1 \).

Key concepts

Convergence

Convergence is a fundamental concept when dealing with series and sequences in mathematics. It refers to the idea that as the number of terms increases towards infinity, the series settles towards a specific limit or sum, rather than diverging.
Essentially, if a series converges, the sequence of partial sums of the series approaches a finite value. This concept is crucial because it helps determine the behavior of functions and the solutions of an infinite series.

• When applying tests for convergence, like the Ratio Test, you are checking if a specific condition holds, ensuring that the infinite series will indeed converge.
• In the context of infinite series, knowing whether a series converges or diverges is critical to understanding its properties and applications.

Understanding convergence helps you determine if and how you can safely sum an infinite series and use it in calculations.

Exponential Series

The exponential series is a particular type of infinite series that can represent functions, especially in calculus and complex analysis. One of the most famous exponential series representations is for the exponential function itself, given by:
\[ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} \]
This series is notable because it converges for all real numbers, an important property for its application in calculus.

• In the given exercise, the series used is a form of the exponential series, but it starts from k=1, which slightly modifies its behavior without affecting its convergence.
• The exponential series converges quickly, making it very useful for approximations in engineering, physics, and other sciences where e^x is frequently encountered.

Remember, the exponential series is a powerful tool in mathematics for both analytical and computational purposes.

Infinite Series

An infinite series is a summation of an infinite sequence of terms. Unlike a finite series, where you can straightforwardly add a finite number of elements, infinite series require special techniques to understand how their sums behave.
Infinite series can either converge, having a sum that approaches some limit, or diverge, meaning they grow indefinitely or oscillate without settling. Important series tests, like the Ratio Test used in the original exercise, help determine convergence.

• The Ratio Test specifically helps identify the values of x for which an infinite series converges by comparing the ratio of successive terms.
• In practical uses, infinite series allow for precise calculations and modeling real-world phenomena, as they can represent complex functions.
  Typical examples include the geometric series and the harmonic series.

Mastering the concept of infinite series is fundamental to advanced mathematics and helps in solving complex problems in calculus and analysis.