Question

The sum of \(\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \dots + \frac{1}{9 \times 10}\) is :

• A. \(\frac{10}{9}\)
• B. \(\frac{9}{10}\)
• C. \(\frac{5}{6}\)
• D. \(\frac{6}{5}\)

Hint:

For any finite telescoping series of the form \(\sum_{n=1}^{N} \frac{1}{n(n+1)}\), the sum is always:
\[ S_N = 1 - \frac{1}{N+1} = \frac{N}{N+1} \]
Here, \(N=9\), so the sum is simply \(\frac{9}{9+1} = \frac{9}{10}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question requires us to find the sum of a finite telescoping series.
In a telescoping series, each term can be split into a difference of two simpler fractions.
When we sum these terms, adjacent terms cancel out, leaving only the first and last terms.
Key Formula or Approach:
The general term of the given series can be represented as:
\[ a_n = \frac{1}{n(n+1)} \]
Using the method of partial fractions, we can write:
\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]

Step 2: Detailed Explanation:
Let us expand each term of the series using the partial fraction decomposition:
- For \(n=1\): \(\frac{1}{1 \times 2} = \frac{1}{1} - \frac{1}{2}\)
- For \(n=2\): \(\frac{1}{2 \times 3} = \frac{1}{2} - \frac{1}{3}\)
- For \(n=3\): \(\frac{1}{3 \times 4} = \frac{1}{3} - \frac{1}{4}\)
This pattern continues up to the last term:
- For \(n=9\): \(\frac{1}{9 \times 10} = \frac{1}{9} - \frac{1}{10}\)
Now, let us sum all these terms together:
\[ S = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \dots + \left(\frac{1}{9} - \frac{1}{10}\right) \]
Notice how the negative term in each bracket is cancelled by the positive term in the next bracket:
- \(-\frac{1}{2}\) cancels with \(+\frac{1}{2}\)
- \(-\frac{1}{3}\) cancels with \(+\frac{1}{3}\)
- And so on, up to \(-\frac{1}{9}\) cancelling with \(+\frac{1}{9}\)
This leaves only the very first term and the very last term:
\[ S = 1 - \frac{1}{10} \]
Subtracting these two values gives:
\[ S = \frac{10 - 1}{10} = \frac{9}{10} \]
Thus, the sum of the series is \(\frac{9}{10}\).

Step 3: Final Answer:
The correct option is (B).