Before You Begin

Question #40672

Difficulty: 8/10 | 🟢 Free

The Question

For each integer $n \ge 1$ , let $F_n(x) = \int_0^x e^{x-t} (t^n - t^{n+1}) dt$ . Define the function $H(x) = \sum_{n=1}^{\infty} F_n(x)$ for $x \in [0, 1]$ . Let $A$ be the left-hand derivative of $H(x)$ at $x=1$ , and $B = \sum_{n=1}^{\infty} F_n'(1)$ . Find the value of $A - B$ .

Options

A.
0
B.
1
C.
e-1
D.
e-2

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

The Telescopic Series Trap|Leibniz Rule for Differentiation under Integral|Mean Value Theorem Applicability

Why This Question Is Tricky

Common Mistakes

This question tests the danger of blindly interchanging differentiation and infinite summation (a common trap when applying the Mean Value Theorem or Taylor expansions near boundaries).

Students often incorrectly assume that $H'(x) = \sum F_n'(x)$ holds uniformly on the closed interval $[0, 1]$ .

However, because the telescopic sum $\sum (x^n - x^{n+1})$ converges non-uniformly at $x=1$ , the derivative of the sum ( $A$ ) and the sum of the derivatives ( $B$ ) diverge exactly at the boundary.

The Leibniz rule correctly identifies the extra boundary terms, exposing the point of non-differentiability of the sum of derivatives.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	The Telescopic Series Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free