How to Calculate the Sum of a Series
Using the Sum of Series Calculator, you can calculate the sum of an infinite series that has a geometric convergence as well as the partial sum of an arithmetic or geometric series. This summation solver can also help you calculate the convergence or divergence of a series.
How Do I Calculate the Sum of an Arithmetic Series?
In an arithmetic series, the difference between each pair of successive terms is constant (called the common difference d). For example, the first 10 odd numbers form an arithmetic series:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
The common difference here is d = 2. Use this formula to find the partial sum:
Sn = n/2 × [2a + (n−1)d]
Where:
- n — Number of terms
- a — First term
- d — Common difference
For the example above (a = 1, d = 2, n = 10):
S10 = 10/2 × [2×1 + (10−1)×2] = 5 × 20 = 100
How Do I Calculate the Sum of a Geometric Series?
In a geometric series, the ratio between each pair of successive terms is constant (called the common ratio r).
To find the partial sum of a geometric series to n terms:
Sn = a × (1 − rn) / (1 − r)
Where:
- a — First term
- r — Common ratio (r ≠ 1)
- n — Number of terms
How Do I Calculate the Sum of an Infinite Geometric Series?
A geometric series can converge or diverge depending on the value of the common ratio r:
- If |r| < 1 → the series converges to a finite sum
- If |r| > 1 → the series diverges (sum is infinite)
- If |r| = 1 → the series is periodic
For a convergent geometric series (|r| < 1), the infinite sum is:
S∞ = a / (1 − r)
Example: For the series 1 + 1/2 + 1/4 + 1/8 + ... (a = 1, r = 0.5):
S∞ = 1 / (1 − 0.5) = 1 / 0.5 = 2
Supported Units
The calculator supports displaying results in:
- Currencies: USD ($), RUB (₽), EUR (€), GBP (£), JPY (¥), CAD, AUD, CHF, CNY, INR, BRL
- Metric units: m, cm, mm, km
- Imperial (US) units: ft, in, yd, mi
FAQs
What is the difference between a series and a sequence?
A sequence is an ordered list of numbers (e.g., 1, 3, 5, 7, …), while a series is the sum of those numbers (e.g., 1 + 3 + 5 + 7 + …).
Can I use this calculator for negative first terms or ratios?
Yes. The calculator handles negative first terms (a), negative common differences (d), and negative common ratios (r), including alternating series where r is between −1 and 0.
What happens if the common ratio r = 1?
When r = 1, the geometric partial sum simplifies to Sn = n × a. The series is periodic and has no finite infinite sum.
Can this calculator handle very large numbers of terms?
Yes, the calculator computes the exact mathematical formula, so even n = 1,000,000 terms produces an accurate result instantly.