Math Problem Statement

Determine whether the series is convergent or divergent by expressing sn as a telescoping sum. If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.) ∞ 21 n(n + 3)

n = 1

Solution

We are given the infinite series:

S=∑n=1∞21n(n+3)S = \sum_{n=1}^{\infty} \frac{21}{n(n+3)}

Step 1: Partial fraction decomposition

We want to decompose 21n(n+3)\frac{21}{n(n+3)} into partial fractions. We express it as:

21n(n+3)=An+Bn+3\frac{21}{n(n+3)} = \frac{A}{n} + \frac{B}{n+3}

Multiplying both sides by n(n+3)n(n+3), we get:

21=A(n+3)+Bn21 = A(n+3) + Bn

Expanding and simplifying

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Math Problem Analysis

Mathematical Concepts

Series
Telescoping Series
Partial Fraction Decomposition

Formulas

-

Theorems

-

Suitable Grade Level

Advanced High School

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