How Do You Find the Sum of an Infinite Number of Terms?

You can find the sum of an infinite number of terms only when the series is a convergent geometric series.

Here's a breakdown:

Geometric Series Explained

A geometric series is a series where each term is multiplied by a constant value (the common ratio, r) to get the next term. The general form looks like this:

a + ar + ar² + ar³ + ...

where:

• a is the first term.
• r is the common ratio.

Convergence is Key

For an infinite geometric series to have a finite sum (i.e., to converge), the absolute value of the common ratio must be less than 1:

|r| < 1

If |r| ≥ 1, the series diverges and does not have a finite sum. The terms either stay the same size (r=1), grow larger (r>1), or oscillate without approaching a limit, making it impossible to calculate a finite sum.

The Formula for the Sum of a Convergent Infinite Geometric Series

If the series converges (|r| < 1), the sum (S) can be calculated using the following formula:

S = a / (1 - r)

where:

• S is the sum of the infinite series.
• a is the first term of the series.
• r is the common ratio.

Steps to Calculate the Sum

1. Identify the first term (a): This is the first number in the series.
2. Calculate the common ratio (r): Divide any term by the preceding term. For example, divide the second term by the first term.
3. Check for convergence: Make sure that the absolute value of r is less than 1 (|r| < 1). If it is not, the series diverges, and you cannot use the formula.
4. Apply the formula: Plug a and r into the formula S = a / (1 - r) to find the sum.

Example

Consider the series: 1 + 1/2 + 1/4 + 1/8 + ...

1. a = 1 (the first term)
2. r = 1/2 ( (1/2) / 1 = 1/2, (1/4) / (1/2) = 1/2, etc.)
3. |r| = |1/2| = 1/2 < 1 (The series converges)
4. S = 1 / (1 - 1/2) = 1 / (1/2) = 2

Therefore, the sum of the infinite series 1 + 1/2 + 1/4 + 1/8 + ... is 2.

Beyond Geometric Series

While the a/(1-r) formula works only for convergent geometric series, other series can also converge, but calculating their sum often requires more advanced techniques from calculus, such as integral tests, ratio tests, or root tests, to determine convergence and different formulas or methods to compute the actual sum. These are beyond the scope of the simple question asked.

In summary, finding the sum of an infinite number of terms is only possible when the series converges, and for geometric series, this convergence depends on the common ratio. When convergent, the formula S = a / (1 - r) provides the sum.