Find The Sum Of The Convergent Series Calculator

This calculator finds the sum of a convergent geometric series, both to infinity (if applicable) and the partial sum of the first 'n' terms. Enter the first term (a), common ratio (r), and number of terms (n) for the partial sum.

Chart of Partial Sums vs. Number of Terms

Table of Terms and Partial Sums

Term (k)	Term Value (ar^k-1)	Partial Sum (Sk)
Enter values to populate table.

What is a Sum of Convergent Series Calculator?

A sum of convergent series calculator, specifically for geometric series, is a tool designed to find the sum of the terms in a geometric sequence that is known to converge. A geometric series converges if its common ratio 'r' has an absolute value less than 1 (i.e., -1 < r < 1). When a geometric series converges, its sum approaches a finite limit as the number of terms increases towards infinity. This calculator helps you find both the sum to infinity (S_∞) and the partial sum of the first 'n' terms (S_n).

This calculator is useful for students studying sequences and series, mathematicians, engineers, and anyone dealing with processes that can be modeled by a geometric progression with a decreasing magnitude of terms. Common misconceptions include thinking all infinite series have an infinite sum; however, convergent series have a finite sum.

Sum of Convergent Geometric Series Formula and Mathematical Explanation

A geometric series is defined by its first term 'a' and a constant common ratio 'r' between successive terms. The terms are a, ar, ar², ar³, …

1. Partial Sum (S_n): The sum of the first 'n' terms of a geometric series is given by:

S_n = a + ar + ar² + … + ar^n-1

To derive the formula, multiply S_n by r:

rS_n = ar + ar² + ar³ + … + arⁿ

Subtracting the second equation from the first:

S_n – rS_n = a – arⁿ

S_n(1 – r) = a(1 – rⁿ)

So, S_n = a(1 – rⁿ) / (1 – r) (for r ≠ 1)

2. Sum to Infinity (S_∞): If the series converges, meaning -1 < r < 1, then as n approaches infinity (n → ∞), rⁿ approaches 0 (rⁿ → 0).

Taking the limit of S_n as n → ∞:

S_∞ = lim_n→∞ [a(1 – rⁿ) / (1 – r)] = a(1 – 0) / (1 – r) = a / (1 – r)

This is the sum to infinity for a convergent geometric series.

Variables Table

Variable	Meaning	Unit	Typical Range
a	First term	Unitless (or units of the context)	Any real number
r	Common ratio	Unitless	-1 < r < 1 for convergence to infinity
n	Number of terms for partial sum	Count	Positive integers (1, 2, 3, …)
Sn	Partial sum of first n terms	Same as 'a'	Depends on a, r, n
S∞	Sum to infinity	Same as 'a'	Finite if -1 < r < 1

Practical Examples (Real-World Use Cases)

Example 1: Repeating Decimal

Consider the repeating decimal 0.3333… This can be written as a geometric series:

0.3 + 0.03 + 0.003 + … = 3/10 + 3/100 + 3/1000 + …

Here, the first term a = 3/10 and the common ratio r = 1/10. Since |r| = 1/10 < 1, the series converges.

Using the sum of convergent series calculator (or formula S_∞ = a / (1 – r)):

S_∞ = (3/10) / (1 – 1/10) = (3/10) / (9/10) = 3/9 = 1/3.

So, 0.3333… = 1/3.

Example 2: Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 60% (0.6) of its previous height. What is the total vertical distance traveled by the ball until it comes to rest?

The distance traveled downwards is 10 + 10(0.6) + 10(0.6)² + …

The distance traveled upwards is 10(0.6) + 10(0.6)² + …

Total downward distance: a = 10, r = 0.6. S_∞,down = 10 / (1 – 0.6) = 10 / 0.4 = 25 meters.

Total upward distance: a = 10(0.6) = 6, r = 0.6. S_∞,up = 6 / (1 – 0.6) = 6 / 0.4 = 15 meters.

Total distance = 25 + 15 = 40 meters. Alternatively, it's 10 (initial drop) + 2 * (sum of upward bounces) = 10 + 2 * 15 = 40m. The sum of convergent series calculator helps here.

How to Use This Sum of Convergent Series Calculator

1. Enter the First Term (a): Input the initial value of your geometric series.
2. Enter the Common Ratio (r): Input the ratio between consecutive terms. For the sum to infinity to be finite, 'r' must be between -1 and 1 (exclusive). The calculator will indicate if the series diverges based on 'r'.
3. Enter the Number of Terms (n): Input the number of terms you want to sum for the partial sum S_n. This must be a positive integer.
4. Calculate: Click "Calculate Sums" or simply change input values. The results will update automatically.
5. Read Results:
   • Primary Result: Shows the sum to infinity (S_∞) if -1 < r < 1, or indicates divergence.
   • Intermediate Results: Displays the partial sum (S_n), rⁿ, 1-r, and convergence status.
   • Chart and Table: Visualize the partial sums and see individual term values.
6. Reset: Click "Reset" to return to default values.
7. Copy Results: Click "Copy Results" to copy the main outputs to your clipboard.

The sum of convergent series calculator provides immediate feedback on convergence and the sums.

Key Factors That Affect Sum of Convergent Series Results

1. First Term (a): The sum is directly proportional to 'a'. If you double 'a', both S_n and S_∞ will double.
2. Common Ratio (r): This is the most critical factor.
   • If |r| < 1, the series converges, and S_∞ is finite. The closer |r| is to 0, the faster it converges.
   • If |r| ≥ 1, the series diverges (unless a=0), and S_∞ is not a finite value (or oscillates). The sum of convergent series calculator handles this.
3. Magnitude of r close to 1: When |r| is less than 1 but very close to 1, the sum to infinity can be very large, and the convergence is slow (many terms needed for S_n to approach S_∞).
4. Sign of r: If r is positive, all terms have the same sign as 'a', and the partial sums monotonically approach S_∞. If r is negative, the terms alternate in sign, and the partial sums oscillate around S_∞ as they converge.
5. Number of Terms (n) for Partial Sum: For a convergent series, as 'n' increases, S_n gets closer to S_∞.
6. Initial Conditions: The starting point 'a' scales the entire series and its sum.

Understanding these factors is crucial when using a sum of convergent series calculator.

Frequently Asked Questions (FAQ)

What if the common ratio |r| is greater than or equal to 1?

If |r| ≥ 1 (and a ≠ 0), the geometric series diverges. This means the partial sums S_n do not approach a finite limit as n increases. The sum to infinity is not finite or well-defined (it goes to ∞, -∞, or oscillates). Our sum of convergent series calculator will indicate this.

Can this calculator handle other types of series?

No, this calculator is specifically designed for geometric series. Other series (like arithmetic, p-series, or more complex ones) require different methods to find their sum or test for convergence. See our series convergence tests page for more.

What does it mean for a series to converge?

A series converges if the sequence of its partial sums (S₁, S₂, S₃, …) approaches a finite limit as the number of terms goes to infinity. For a geometric series, this happens when -1 < r < 1.

How quickly does a geometric series converge?

The speed of convergence depends on the absolute value of 'r'. The smaller |r| is, the faster rⁿ goes to zero, and the faster S_n approaches S_∞. You can see this in the chart generated by the sum of convergent series calculator.

What if the first term 'a' is zero?

If a=0, then all terms are zero, and the sum (both partial and to infinity) is zero, regardless of 'r'.

Can 'r' be negative?

Yes, if 'r' is negative (and |r| < 1), the series is an alternating geometric series, and it still converges. The partial sums will oscillate around the sum to infinity.

Why is the formula S_n = a(1 – rⁿ) / (1 – r) not used when r=1?

If r=1, the denominator becomes zero, and the formula is undefined. If r=1, the series is a + a + a + …, which is na. If a≠0, this diverges to infinity.

Is the sum to infinity an exact value or an approximation?

For a convergent geometric series, the sum to infinity S_∞ = a / (1 – r) is an exact value, representing the limit of the partial sums.