Find 10 Partial Sums of the Series and Round Your Answers to Five Decimal Places

To find the partial sums of a given series, we start by determining the first ten terms of the series and then add them up incrementally. Let’s say the series is represented as follows:

• S = a₁ + a₂ + a₃ + … + a_n

To find each partial sum, we calculate:

• S₁ = a₁
• S₂ = a₁ + a₂
• S₃ = a₁ + a₂ + a₃
• … and so on until S₁₀.

Once we have all the sums calculated, we round each of the results to five decimal places.

For example, let’s consider the series where the terms are defined as:

a_n = rac{1}{n}

This means:

• a₁ = 1
• a₂ = 0.5
• a₃ = 0.33333
• a₄ = 0.25
• a₅ = 0.2
• a₆ = 0.16667
• a₇ = 0.14286
• a₈ = 0.125
• a₉ = 0.11111
• a₁₀ = 0.1

The first ten partial sums would be calculated as:

• S₁ = 1 = 1.00000
• S₂ = 1 + 0.5 = 1.50000
• S₃ = 1 + 0.5 + 0.33333 = 1.83333
• S₄ = 1 + 0.5 + 0.33333 + 0.25 = 2.08333
• S₅ = 1 + 0.5 + 0.33333 + 0.25 + 0.2 = 2.28333
• S₆ = 1 + 0.5 + 0.33333 + 0.25 + 0.2 + 0.16667 = 2.45000
• S₇ = 1 + 0.5 + 0.33333 + 0.25 + 0.2 + 0.16667 + 0.14286 = 2.59286
• S₈ = 1 + 0.5 + 0.33333 + 0.25 + 0.2 + 0.16667 + 0.14286 + 0.125 = 2.71786
• S₉ = 1 + 0.5 + 0.33333 + 0.25 + 0.2 + 0.16667 + 0.14286 + 0.125 + 0.11111 = 2.82897
• S₁₀ = 1 + 0.5 + 0.33333 + 0.25 + 0.2 + 0.16667 + 0.14286 + 0.125 + 0.11111 + 0.1 = 2.92997

The final rounded results for the first ten partial sums of the series are:

1. 1.00000
2. 1.50000
3. 1.83333
4. 2.08333
5. 2.28333
6. 2.45000
7. 2.59286
8. 2.71786
9. 2.82897
10. 2.92997

Thus, these are the ten partial sums rounded to five decimal places.