Problem 294 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find \(\sum_{n=1}^{\infty} \frac{1}{(n+1) !}\)

Step-by-Step Solution

Answer

The sum of the series is \(e - 1\)

Step 1: Understanding Factorials

For any natural number \(n\), the factorial \(n!\) is the product of all positive integers less than or equal to \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1\). The factorial function has special cases, \(0! = 1\) and \(1! = 1\). It's also important to know that \((n+1)! = (n+1) \times n!\) which simplifies computations tremendously.

Step 2: Rewriting the series in recognizable form

Rewrite the given series \(\sum_{n=1}^{\infty} \frac{1}{(n+1)!}\) as \(\sum_{n=1}^{\infty} \frac{1}{n!(n+1)}\). Upon rewriting, the terms start resembling the well-known series for the exponential function's power series representation \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\). By shifting \(n\) to \(n+1\), we get \(\sum_{n=1}^{\infty} \frac{1}{n!(n)} = \sum_{n=0}^{\infty} \frac{1}{(n+1)!} = e - 1\). The subtraction of 1 accounts for the starting index difference.

Step 3: Finding the Sum

The sum of the given series equals \(e - 1\) where \(e\) is the base of natural logarithm approximately equal to 2.71828.

Key Concepts

Factorial

A factorial, symbolized as "!", is a mathematical operation applied to a positive integer. It involves multiplying the number by each positive whole number less than itself. For example:

• For 5, it is calculated as: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

One of the critical properties of the factorial is its recursive nature: \((n+1)! = (n+1) \times n!\).This property is particularly useful in simplifying expressions involving factorials, like in many series problems. Factorials have specific values for small numbers as well, such as:

• \(0! = 1\) and \(1! = 1\).

These special cases are vital to grasp when working with series involving factorials, allowing simpler computations and transformations in summation problems.

Exponential Function

The exponential function is one of the most important functions in mathematics, represented as \(e^x\).Here, "e" is Euler's number, and it is an irrational constant approximately equal to 2.71828. The exponential function is unique in that its rate of growth is proportional to its size—it's continuously rising and appears in many natural phenomena.

• It's used to model compound growth, such as populations or investments.
• It shows up in differential equations that describe natural processes.

In terms of series, the exponential function can also be represented as an infinite series with the formula:\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]This power series representation is crucial for calculations in calculus and analysis, enabling mathematicians to approximate values or solve equations involving exponentials with greater precision.

Power Series Representation

Power series are infinite series used to express functions as sums:\(f(x) = \sum_{n=0}^{\infty} a_n x^n\),where \(a_n\) represents the series coefficients and \(x^n\) the power terms. Power series are vital in mathematics as they allow complex functions to be manipulated, explored, and understood through their simpler polynomial components.

• The exponential function is a classic example, expanded into \(\sum_{n=0}^{\infty} \frac{x^n}{n!}\), giving powerful insights into its behavior.
• The series converges for all real values of \(x\), which is one of the reasons it gets such widespread use in mathematical modeling.

When assessing series like \(\sum_{n=1}^{\infty} \frac{1}{(n+1)!}\), rewriting them to resemble a known power series helps identify sums and behavior. This method involves shifting indices or changing terms to fit into known formats, thus making complex series easier to understand and evaluate.