Evaluate the Summation sum from n=1 to 8 of 1/n

Problem

(∑_n=1^8)(1/n)

Solution

1. Identify the terms of the finite harmonic series by substituting each integer value from n=1 to n=8 into the expression 1/n

2. Write out the individual fractions for the sum.

1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8

3. Find a common denominator to add the fractions. The least common multiple of the denominators {1,2,3,4,5,6,7,8} is 840

4. Convert each fraction to have the denominator 840

840/840+420/840+280/840+210/840+168/840+140/840+120/840+105/840

5. Sum the numerators together.

840+420+280+210+168+140+120+105=2283

6. Simplify the resulting fraction 2283/840 by dividing both the numerator and denominator by their greatest common divisor, which is 3

(2283÷3)/(840÷3)=761/280

Final Answer

(∑_n=1^8)(1/n)=761/280

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