How to Find the nth Partial Sum

The nth partial sum of a series is the sum of its first n terms. Here's how to find it:

Understanding Partial Sums

A partial sum represents a finite sum of terms from a potentially infinite series. Specifically, the nth partial sum, denoted as S_n, is the sum of the first n terms of a sequence.

Formula for the nth Partial Sum

The nth partial sum (S_n) of a series ∑ k = 1 ∞ a k is defined as:

S_n = ∑ k = 1 n a k = a₁ + a₂ + a₃ + ... + a_n

Where:

• S_n is the nth partial sum.
• a_k represents the kth term of the sequence.
• n is the number of terms being summed.

Steps to Find the nth Partial Sum

1. Identify the Series: Determine the series you're working with and understand how the terms are generated.
2. Determine 'n': Identify the value of 'n', which represents the number of terms you need to sum. This is usually specified in the problem (e.g., "Find the 5th partial sum").
3. Calculate the First 'n' Terms: Calculate the values of the first 'n' terms of the series (a₁, a₂, a₃, ..., a_n).
4. Sum the Terms: Add the first 'n' terms together. The result is the nth partial sum (S_n).

Examples

Example 1: Arithmetic Series

Find the 4th partial sum of the arithmetic series: 2 + 4 + 6 + 8 + ...

• n = 4
• a₁ = 2, a₂ = 4, a₃ = 6, a₄ = 8
• S₄ = 2 + 4 + 6 + 8 = 20

Therefore, the 4th partial sum is 20.

Example 2: Geometric Series

Find the 3rd partial sum of the geometric series: 1 + 1/2 + 1/4 + ...

• n = 3
• a₁ = 1, a₂ = 1/2, a₃ = 1/4
• S₃ = 1 + 1/2 + 1/4 = 7/4

Therefore, the 3rd partial sum is 7/4.

Example 3: Using Sigma Notation

Find the 5th partial sum of the series ∑ k = 1 ∞ k².

• n = 5
• The terms are: 1², 2², 3², 4², 5² which equals 1, 4, 9, 16, 25.
• S₅ = 1 + 4 + 9 + 16 + 25 = 55

Therefore, the 5th partial sum is 55.

Formula Shortcuts

For certain types of series, like arithmetic and geometric series, there are specific formulas to directly calculate the nth partial sum without having to calculate and add all the individual terms:

• Arithmetic Series: S_n = (n/2) * [2a₁ + (n - 1)d], where a₁ is the first term and d is the common difference.
• Geometric Series: S_n = a₁ * (1 - rⁿ) / (1 - r), where a₁ is the first term and r is the common ratio (r ≠ 1).

Using these formulas can save time and reduce the risk of error, especially for larger values of 'n'.

Summary

Finding the nth partial sum involves understanding the series, identifying the value of n, calculating the first n terms, and then summing those terms. For arithmetic and geometric series, specific formulas can streamline the calculation.