Problem 38 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}(\log a)^{n}\), then at \(x=0, f(x)\) (A) has no limit (B) is discontinuous (C) is continuous but not differentiable (D) is differentiable

Step-by-Step Solution

Answer

(D) is differentiable.

Step 1: Understanding the function

The given function is a series, specifically the exponential function. The function is defined as \( f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} (\log a)^n \). This resembles the power series of the exponential function \( e^x \), which is \( \sum_{n=0}^{\infty}\frac{x^n}{n!} \).

Step 2: Analyzing the function at x=0

Let's evaluate the function at \( x = 0 \). Substituting \( x = 0 \) into the series gives \( f(0) = \sum_{n=0}^{\infty} \frac{0^n}{n!} (\log a)^n \). For \( n \geq 1 \), \( 0^n = 0 \), hence every term is zero except when \( n = 0 \). Here, it simplifies to \( \frac{(\log a)^0}{0!} = 1 \). Thus, \( f(0) = 1 \).

Step 3: Determining continuity and differentiability

Since \( f(x) \) is a power series centered at \( x = 0 \), it converges for all real numbers, implying that \( f(x) \) is both continuous and infinitely differentiable everywhere on its interval of convergence. Thus, at \( x = 0 \), \( f(x) \) is not only continuous but also differentiable.

Key Concepts

Exponential Function

The exponential function is a fundamental concept in mathematics and is defined by the formula \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \). This function has significant applications across various scientific fields because of its unique properties. For example, it grows at a rate proportional to its current value.

• One of the remarkable features of the exponential function is its ability to model growth and decay processes, such as population growth or radioactive decay.
• It is also periodic in nature, which means the derivative of \( e^x \) is the function itself.
• The exponential series converges for all real numbers, which makes it extremely useful in solving differential equations and appearing in calculus problems.

In the context of power series, the given function \( f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} (\log a)^n \) is an example where the pattern of the exponential series is applied, but with a slight modification involving the factor \((\log a)^n\). This demonstrates the flexibility and adaptability of such series meanings to represent a vast range of functions.

Continuity

Continuity is a vital property of functions, indicating that a function behaves predictably within its domain. A function is continuous at a point \( x = a \) if the limit as \( x \) approaches \( a \) equals the function's value at \( a \). Simply put, there are no sudden jumps or breaks in the graph of the function at this point.

• Mathematically, a function \( f \) is continuous at \( x = 0 \) if \( \lim_{x \to 0} f(x) = f(0) \).
• For power series, continuity is generally guaranteed within the interval of convergence, which is usually determined by where the series converges.
• In our exercise, the power series representation of \( f(x) \) implies that it is continuous everywhere on its interval of convergence, inclusive of \( x = 0 \).

Hence, the power series being evaluated at \( x = 0 \) yields \( f(0) = 1 \), confirming that the function remains continuous at this point.

Differentiability

Differentiability is a concept that refers to whether a function has a derivative at certain points in its domain. A function \( f(x) \) is considered differentiable at a point \( x = a \) if the derivative \( f'(x) \) exists at that point. A function that is differentiable at \( x = a \) will also be continuous at that point, but the reverse is not necessarily true.

• The derivative of a power series can be found by differentiating each term individually, which is both an exciting and powerful aspect of working with power series.
• Given that the series \( f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} (\log a)^n \) is a power series, differentiability is already assumed wherever the series converges.
• This means that the function is not just continuous at \( x = 0 \), but also differentiable.

Therefore, at \( x = 0 \), our function \( f(x) \) maintains differentiability, demonstrating one of the elegant advantages of power series: their ease in exhibiting both continuity and differentiability at points within their domain.