Math Problem Statement

7 2 × 3 × 4

  • 9 3 × 4 × 5
  • 11 4 × 5 × 6
  • ⋅ ⋅ ⋅ + 2n + 3 n(n + 1)(n + 2).

Solution

The expression you've provided appears to follow a specific pattern. Let's break it down step by step:

The general term in the sum is:

2n+3n(n+1)(n+2)\frac{2n + 3}{n(n + 1)(n + 2)}

You have the sum of the series starting from n=1n = 1:

S=72×3×4+93×4×5+114×5×6++2n+3n(n+1)(n+2)S = \frac{7}{2 \times 3 \times 4} + \frac{9}{3 \times 4 \times 5} + \frac{11}{4 \times 5 \times 6} + \cdots + \frac{2n + 3}{n(n+1)(n+2)}

To simplify and explore the sum further, observe that each term follows this general structure, which simplifies as follows:

Sn=2n+3n(n+1)(n+2)S_n = \frac{2n + 3}{n(n+1)(n+2)}

This can be decomposed using partial fractions:

2n+3n(n+1)(n+2)=An+Bn+1+Cn+2\frac{2n + 3}{n(n+1)(n+2)} = \frac{A}{n} + \frac{B}{n+1} + \frac{C}{n+2}

To determine the values of AA, BB, and CC, solve the system obtained by rewriting the left-hand side in terms of these fractions.

Would you like to go into the details of solving the partial fractions and evaluating the sum for a specific nn?


5 Related Questions:

  1. How do you solve a partial fraction decomposition?
  2. Can you find the closed form for this sum as nn goes to infinity?
  3. How do you analyze the convergence of an infinite series?
  4. How does the general term of a series affect the behavior of the sum?
  5. What are other examples of series that involve rational expressions?

Tip:

Breaking down complex rational expressions using partial fractions can simplify calculations and help identify patterns in sums.

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Math Problem Analysis

Mathematical Concepts

Algebra
Series and Sequences
Partial Fractions

Formulas

General term: (2n + 3) ÷ (n(n+1)(n+2))
Partial fraction decomposition: A ÷ n + B ÷ (n+1) + C ÷ (n+2)

Theorems

Partial Fraction Decomposition
Convergence of Series

Suitable Grade Level

Grades 10-12