Problem 2 - Essential Calculus

Question

Find the first five terms of the sequence of partial sums. $$ \frac{1}{2 \cdot 3}+\frac{2}{3 \cdot 4}+\frac{3}{4 \cdot 5}+\frac{4}{5 \cdot 6}+\frac{5}{6 \cdot 7}+\cdots $$

Step-by-Step Solution

Answer

The first five terms of the sequence are: 0.167, 0.25, 0.3, 0.333, 0.357

Step 1: Identify the pattern

Each term in the infinite series can be described as: \(\frac{n}{(n+1)(n+2)}\) , where \(n\) starts from 1. This common structure provides an insight into the pattern of the sequence.

Step 2: Break down the fraction

We see that each term in the sequence is a fraction. We can break down each term in the following way: \(\frac{n}{(n+1)(n+2)}=\frac{1}{n+1} - \frac{1}{n+2}\)

Step 3: Find the first five terms

Now, compute the first five terms by adding the terms according to the derived formula:- For \(n=1\), the first term is: \(\frac{1}{1+1} - \frac{1}{1+2} = 0.5 - 0.333 = 0.167\)- For \(n=2\), the sum of first two terms is: \(0.167 + (\frac{1}{2+1} - \frac{1}{2+2}) = 0.167 + 0.333 - 0.25 = 0.25\)- For \(n=3\), the sum of first three terms is: \(0.25 + (\frac{1}{3+1} - \frac{1}{3+2}) = 0.25 + 0.25 - 0.2 = 0.3\)- For \(n=4\), the sum of first four terms is: \(0.3 + (\frac{1}{4+1} - \frac{1}{4+2}) = 0.3 + 0.2 - 0.167 = 0.333\)- For \(n=5\), the sum of first five terms is: \(0.333 + (\frac{1}{5+1} - \frac{1}{5+2}) = 0.333 + 0.167 - 0.143 = 0.357\)

Step 4: Write down the sequence

The sequence of the first five partial sums is: 0.167, 0.25, 0.3, 0.333, 0.357