Definite Integration
Limits of Integrals
Grade 12
Question:
<p>Let <span class="math">\(I_n = \int_{-1}^{1} x \left(1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^{2n}}{2n}\right) dx\)</span>. If <span class="math">\(\lim_{n \to \infty} I_n\)</span> can be expressed as <span class="math">\(\frac{p}{q}\)</span> in its lowest form, then find the value of <span class="math">\(\frac{p}{q(p+q)}{10}\)</span>.</p>
Step-by-Step Solution
Key Concept: Recognize the power series expansion and use properties of definite integration combined with limit evaluation.
<p><strong>Solution:</strong> Evaluate the integral and find the limit as <span class="math">$n \to \infty$</span>, then express as a fraction in lowest form and compute the required expression.</p>
Correct Answer: 3