1. What is Partial Sum?

The partial sum (Sₙ) of a sequence is the sum of its first n terms. It's a fundamental concept in series and sequence analysis in mathematics.

2. How Does the Calculator Work?

The calculator uses the partial sum formula:

Where:

• \( S_n \) — Partial sum of first n terms (unitless)
• \( a_k \) — k-th term of the sequence (unitless)
• \( n \) — Number of terms (integer)

Explanation: The calculator computes cumulative sums of the sequence terms, showing both the final result and intermediate partial sums.

3. Importance of Partial Sums

Details: Partial sums are essential for analyzing series convergence, calculating cumulative totals, and understanding growth patterns in sequences.

4. Using the Calculator

Tips: Enter comma-separated numerical values (e.g., "1, 3, 5, 7"). The calculator will compute all partial sums from S₁ to Sₙ.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between partial sum and series sum?
A: Partial sum is the sum of first n terms, while series sum is the limit of partial sums as n approaches infinity (if it exists).

Q2: Can I use fractions or decimals?
A: Yes, the calculator accepts both decimal (e.g., 1.5) and fractional (e.g., 3/4) inputs.

Q3: What if my sequence has a pattern?
A: For patterned sequences, consider entering more terms or using a formula-based approach for infinite series.

Q4: Is there a limit to the number of terms?
A: The calculator can handle hundreds of terms, but extremely long lists may impact performance.

Q5: How are partial sums used in real-world applications?
A: They're used in financial calculations (cumulative interest), physics (discrete systems), and computer science (algorithm analysis).