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Question #41417

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \\sum_{k=1}^{\\infty} \\frac{2}{(x+k)(x+k+1)(x+k+2)}$ . Define $H(x) = \\int_0^x f(t) dt - \\int_0^x \\frac{1}{2(1+t^3)} dt$ . The number of positive real roots of the equation $H(x) = 0$ is:

Options

A.
0
B.
1
C.
2
D.
3

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

Partial Fractions Decomposition|Method of Differences|Inequalities in Definite Integrals|Rolle's Theorem Root Existence

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to synthesize partial fraction telescoping series with integral evaluation and functional analysis.

A common trap (Trap 287) is failing to correctly evaluate the limit of the telescoping sum, leaving residual terms.

Furthermore, students often overlook the monotonicity shift (Trap 255) and assume that since $H(0)=0$ and $H'(3/2)=0$ , no further roots exist, failing to check the behavior at infinity.

The 'Lethal Collision' occurs when evaluating the improper integral $\\int_0^\\infty \\frac{1}{1+t^3} dt$ using the $t=1/u$ substitution to bound the function's asymptotic behavior, forcing a rigorous application of the Intermediate Value Theorem and Rolle's Theorem to prove the existence of exactly one positive root.

Quick Facts About This Question

Question Information

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