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Find the sequence of partial sums \(s_1, s_2, s_3, s_4, \) and \(s_5\). (Round your answers to four decimal places.)

\[ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \ldots \]

Answer :

Final answer:

The question deals with finding partial sums of a series, which is a concept in mathematics. The partial sums for the first five terms of the series provided are found to be 1, 1.25, 1.3611, 2.3611, and 2.4011 respectively.

Explanation:

The given sequence is a series that represents the sum of the reciprocals of perfect squares. The general term of the series is 1/n², where n represents the position in the sequence.

To find the partial sums (s₁, s₂, s₃, s₄, and s₅), we can calculate the cumulative sum up to each term:

1. s₁ = 1

2. s₂ = 1 + 1/4

3. s₃ = 1 + 1/4 + 1/9

4. s₄ = 1 + 1/4 + 1/9 + 1/16

5. s₅ = 1 + 1/4 + 1/9 + 1/16 + 1/25

Let's compute these:

1. s₁ = 1

2. s₂ = 1 + 0.25 = 1.25

3. s₃ = 1 + 0.25 + 0.1111 = 1.3611

4. s₄ = 1 + 0.25 + 0.1111 + 0.0625 = 1.4236

5. s₅ = 1 + 0.25 + 0.1111 + 0.0625 + 0.04 = 1.4636

So, the rounded partial sums to four decimal places are s₁ = 1, s₂ = 1.25, s₃ = 1.3611, s₄ = 1.4236, and s₅ = 1.4636. If you have further questions or need clarification, feel free to ask!

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