Math Problem Statement

find a formula for the nth partial sum then find the sum if it converges

Solution

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Math Problem Analysis

Mathematical Concepts

Series and Sequences
Partial Fractions
Telescoping Series
Limits

Formulas

General Term: \( a_n = \frac{6}{n(n+1)} \)
Partial Fraction Decomposition: \( \frac{6}{n(n+1)} = \frac{6}{n} - \frac{6}{n+1} \)
Partial Sum Formula: \( S_n = 6 - \frac{6}{n+1} \)
Limit of a Series: \( \lim_{n \to \infty} S_n = 6 \)

Theorems

Limit Theorem for Series
Properties of Telescoping Series

Suitable Grade Level

Undergraduate Mathematics or Advanced High School (Grades 11-12)