Partial Sum Calculator

What Is a Partial Sum?

A partial sum is the sum of the first n terms of a sequence. It gives a way to approximate infinite series by summing a finite number of initial terms. Partial sums are used in mathematics, physics, and economics to understand series behavior and convergence.

Types of Series You Can Compute

• ➕ Arithmetic series: sums with a constant difference between terms
• ✖️ Geometric series: sums with a constant ratio between terms
• 🔢 Custom numerical series
• ∑ Sigma notation inputs

How to Use the Partial Sum Calculator

1. Choose the type of series: arithmetic, geometric, or custom
2. Enter the first term, common difference or ratio, and number of terms (n)
3. If using sigma notation, input the general term and bounds
4. Click "Calculate" to get the partial sum
5. View the result with steps and relevant formula used

Partial Sum Examples

• Example 1 (Arithmetic): \( a = 2,\ d = 3,\ n = 5 \) → \( S_5 = \frac{5}{2} \times (2\times2 + (5 - 1)\times3) = 40 \)
• Example 2 (Geometric): \( a = 1,\ r = 2,\ n = 4 \) → \( S_4 = \frac{1(1 - 2^4)}{1 - 2} = 15 \)
• Example 3 (Sigma Notation): \( \sum_{k=1}^{5} (2k + 1) \) → Result: 35

Applications of Partial Sums

• 📈 Approximating infinite series
• 📘 Solving math and calculus problems
• 💰 Analyzing financial models
• 📐 Understanding convergence in sequences
• 🔬 Estimating outcomes in physical sciences

Frequently Asked Questions (FAQ)