Summing Fractional Series: A Comprehensive Guide
Understanding how to sum up a series of fractions, especially when dealing with different denominators, is a fundamental concept in mathematics. This guide will walk you through the process of finding the sum of the series 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10. We'll cover both the step-by-step approximation method and the exact method using a common denominator.
Step-by-Step Approximation
Let's start by approximating the sum of the given series:
Step 1: Convert Each Fraction to a Decimal Value
Convert each fraction in the series to its decimal form:
1/2 0.5 1/3 ≈ 0.3333 1/4 0.25 1/5 0.2 1/6 ≈ 0.1667 1/7 ≈ 0.1429 1/8 0.125 1/9 ≈ 0.1111 1/10 0.1Step 2: Add the Decimal Values
Add the collected decimal values together in a step-by-step manner:
0.5 0.3333 0.8333 0.8333 0.25 1.0833 1.0833 0.2 1.2833 1.2833 0.1667 ≈ 1.4500 1.4500 0.1429 ≈ 1.5929 1.5929 0.125 1.7179 1.7179 0.1111 ≈ 1.8290 1.8290 0.1 1.9290This gives us an approximate sum of the series: 1.929.
Exact Sum Using a Common Denominator
For a more precise calculation, we can find the exact sum using a common denominator for all fractions. The least common multiple (LCM) of the denominators 2, 3, 4, 5, 6, 7, 8, 9, 10 is 2520.
Step 1: Convert Each Fraction with a Common Denominator
Express each term with a denominator of 2520:
1/2 1260/2520 1/3 840/2520 1/4 630/2520 1/5 504/2520 1/6 420/2520 1/7 360/2520 1/8 315/2520 1/9 280/2520 1/10 252/2520Step 2: Add the Fractions
Add the fractions together:
1260 840 630 504 420 360 315 280 252 2520 4571 2520
This gives us the exact sum: 4571/2520 ≈ 1.813.
Alternative Method: A Simplified Sum
Another method to sum the series is by employing a different approach. For 1/2, 1/3, 1/4, 1/5, we can calculate the sum directly:
1/2 1/3 1/4 1/5We can convert it to a simpler form:
1/2 1/3 1/4 1/5 12/24 8/24 6/24 4.8/24 12 8 6 4.8 26/24 26/24 13/12 13/12 1 1/12This provides a straightforward and easy-to-understand sum of the series: 1 and 1/12.
Conclusion
Understanding the sum of a series of fractions is crucial in various mathematical applications. This guide has provided detailed steps for approximating and finding the exact sum of a fractional series, offering clarity and precision in mathematical calculations.