Sum of The Following Series Calculator

How to Use This Calculator

Using our sum of the following series calculator is simple. Follow these steps:

1. Select the type of series you want to calculate (arithmetic, geometric, or other).
2. Enter the required values in the input fields. For arithmetic series, you'll need the first term, common difference, and number of terms. For geometric series, you'll need the first term, common ratio, and number of terms.
3. Click the "Calculate" button to get the sum of the series.
4. Review the result and use the chart to visualize the series if needed.

The calculator will display the sum of the series and provide a step-by-step explanation of how the calculation was performed.

Formula for Sum of Series

The formula used to calculate the sum of a series depends on the type of series. Here are the formulas for common types:

For other types of series, different formulas may apply. The calculator will use the appropriate formula based on the series type you select.

Worked Examples

Let's look at some examples to understand how the sum of a series is calculated.

Example 1: Arithmetic Series

Find the sum of the first 10 terms of an arithmetic series where the first term is 2 and the common difference is 3.

Using the formula for an arithmetic series:

S = n/2 × (2a + (n-1)d)

Plugging in the values:

S = 10/2 × (2×2 + (10-1)×3) = 5 × (4 + 27) = 5 × 31 = 155

The sum of the series is 155.

Example 2: Geometric Series

Find the sum of the first 5 terms of a geometric series where the first term is 3 and the common ratio is 2.

Using the formula for a geometric series:

S = a × (1 - rⁿ) / (1 - r)

Plugging in the values:

S = 3 × (1 - 2⁵) / (1 - 2) = 3 × (1 - 32) / (-1) = 3 × (-31) / (-1) = 93

The sum of the series is 93.

Types of Series

There are several types of series that you may encounter. Here are some common types:

Arithmetic Series

An arithmetic series is a sequence of numbers where the difference between consecutive terms is constant. The sum of an arithmetic series can be calculated using the formula mentioned earlier.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. The sum of a geometric series can be calculated using the formula mentioned earlier.

Harmonic Series

A harmonic series is a series of numbers where each term is the reciprocal of an integer. The sum of the first n terms of a harmonic series is given by the nth harmonic number, H_n.

Power Series

A power series is a series of the form Σa_nxⁿ, where a_n and x are constants. Power series are used in many areas of mathematics and physics.