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Question #41441

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_n = \int_0^1 x^n e^{x^2} \, dx$ for $n \in \mathbb{N} \cup \{0\}$ . Find the value of $\sum_{n=1}^{50} \left[ (2n-1) I_{2n-2} \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
97
B.
98
C.
99
D.
100

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Reduction Formulae|Sum to n terms vs. General Term|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

Students often assume that the limit of a sequence can be used to evaluate the floor of its terms for all large n, ignoring the discrete jumps.

Here, as n approaches infinity, the integral term approaches 0, so the sequence term approaches e.

Students might blindly assume the floor is [e] = 2 for all n, yielding 50 × 2 = 100.

The 'Lethal Collision' forces them to manually check the first few terms using Taylor expansion to find exactly where the sequence crosses the integer threshold, penalizing over-reliance on asymptotic limits.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Reduction Formulae
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free