Problem 35 - Essential Calculus

Question

In Exercises \(35-38,\) write an equivalent series with the index of summation beginning at \(n=1\). $$ \sum_{n=0}^{\infty} \frac{x^{n}}{n !} $$

Step-by-Step Solution

Answer

The equivalent series of the given series with the index of summation beginning at \(n=1\) is \(\sum_{n=1}^{\infty} \frac{x^n/n}{n !}\).

Step 1: Write down the given series

First, write down the given series \(\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\). The index of series in this series starts at 0. Our aim is to convert this into a series where index starts at 1.

Step 2: Substitute \(n\) with \(n-1\)

The trick to write an equivalent series with the index starting from \(n=1\) is to substitute \(n\) with \(n-1\) in the given series. This will not change the value of the series but the index starting point will shift by 1. So, in place of \(n\), we substitute \(n-1\), and the series becomes \(\sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1) !}\).

Step 3: Simplify the series

Simplify the series by multiplying the numerator and denominator by \(n\), such that \(x^{n-1} = x^n/x\) and \((n-1)! = n!/n\). This gives us the equivalent series as \(\sum_{n=1}^{\infty} \frac{x^n/n}{n !}\).

Key Concepts

Index of Summation

When dealing with a mathematical series, each term is defined by an index, often represented by the variable \( n \). The index of summation tells us where to begin and is key to understanding how a series behaves.
In the original problem, the series starts with an index \( n=0 \). Sometimes it's helpful to shift this starting point to \( n=1 \), particularly when integrating the series into broader mathematical concepts or standard forms.
To shift the index, you substitute \( n \) with a new variable that accounts for the change. In this case, we use \( n-1 \), which effectively starts the sequence from \( n=1 \). This technique is crucial because it enables mathematical manipulation without altering the value of the series.

• Shifting indices requires consistency; you must adjust all terms dependent on \( n \) to reflect the new index.
• The algebraic expressions depend on the initialization point, so it's important to properly change it during conversions.

Understanding index manipulation can make recognizing and simplifying series much smoother and allows for flexible procedural adjustments.

Mathematical Series

A mathematical series is essentially a sum of terms. This can be expressed in the form \(\sum_{n=a}^{b} \). Each term typically follows a specific pattern or rule defined by a formula in terms of \( n \). For infinite series, as seen in the exercise, \( b \) is infinity, which means the series continues indefinitely.
When manipulating series, like the one we're working with, series equivalence is a significant concept. Though the sequence of terms may shift, their collective sum remains unchallenged. This stability allows transformations such as changing the index without altering the outcome.
By handling these series, you're often simplifying expressions or converting them to a more feasible form for solving equations or integrating.

• Equivalence transformations typically involve algebraic manipulations that retain the sum's value.
• These transformations simplify calculations or make mathematical models more consistent with desired forms.

Recognizing these patterns greatly aids in deeper comprehension of complex mathematical concepts.

Factorial in Series

Factorials, denoted by \( n! \), mean multiplying a series of descending natural numbers down to 1. For instance, \( 3! = 3 \times 2 \times 1 = 6 \). They provide an essential function within many mathematical series, especially when terms grow or shrink factorially.
In the original series, each term features a factorial in the denominator \(\frac{1}{n!}\). This factorial manages the growth of the series, ensuring each succeeding term contributes decreasingly less impact towards the sum if \( n \) goes towards infinity.
Factorials ensure convergence in series like the exponential function, balancing the exponential growth in the numerator. They are crucial for stability in series calculations, limiting extreme results.

• The factorial's role usually governs how rapidly terms thin out in relation to the series, affecting convergence or divergence.
• In transformations, adjustments to factorials ensure the mathematical integrity of a series.

Understanding factorials within series strengthens one’s grasp on how summations behave insomuch as growth and balance are concerned.