Calculate The Sum of Convergent Series 4 N 2

The sum of the series 4^n / 2^n is a fundamental mathematical problem that appears in various fields including computer science, physics, and engineering. This calculator helps you compute the sum of the series for any given number of terms.

What is the sum of the series 4^n / 2^n?

The series 4^n / 2^n is a geometric series where each term is a fraction of powers of 4 and 2. A geometric series has the general form:

General Geometric Series

S = a + ar + ar² + ar³ + ... + ar^(n-1)

For the series 4^n / 2^n, we can rewrite it as (2²)^n / 2^n = 2^(2n) / 2^n = 2^n. So the series simplifies to 2 + 2² + 2³ + ... + 2^n.

This is a geometric series with first term a = 2 and common ratio r = 2. The sum of a finite geometric series is given by:

Sum of Finite Geometric Series

S = a(1 - r^n) / (1 - r)

For our series, substituting a = 2 and r = 2 gives:

Sum of Series 4^n / 2^n

S = 2(1 - 2^n) / (1 - 2) = 2(1 - 2^n) / -1 = 2(2^n - 1)

Formula for the series sum

The sum of the series 4^n / 2^n for n terms can be calculated using the formula:

Sum Formula

S = 2(2^n - 1)

Where:

S is the sum of the series
n is the number of terms in the series

This formula is derived from the properties of geometric series and the simplification of the original series terms.

How to calculate the sum

To calculate the sum of the series 4^n / 2^n:

Identify the number of terms (n) in the series.
Apply the formula S = 2(2^n - 1).
Compute the result using the formula.

The calculator on this page automates these steps for you. Simply enter the number of terms and click "Calculate" to get the sum.

Worked examples

Example 1: n = 3

The series is: 4¹/2¹ + 4²/2² + 4³/2³ = 2 + 4 + 8 = 14

Using the formula: S = 2(2³ - 1) = 2(8 - 1) = 2 × 7 = 14

Example 2: n = 5

The series is: 2 + 4 + 8 + 16 + 32 = 62

Using the formula: S = 2(2⁵ - 1) = 2(32 - 1) = 2 × 31 = 62

Example 3: n = 10

The series is: 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 = 2046

Using the formula: S = 2(2¹⁰ - 1) = 2(1024 - 1) = 2 × 1023 = 2046

FAQ

What is the difference between the series 4^n / 2^n and 2^n?

The series 4^n / 2^n simplifies to 2^n, so both represent the same series of powers of 2. The original form is just a different way of writing the same terms.

Is the series 4^n / 2^n convergent?

Yes, the series 4^n / 2^n is convergent because it simplifies to the geometric series 2^n, which converges as n approaches infinity.

Can I use this formula for negative values of n?

No, the formula S = 2(2^n - 1) is valid only for positive integer values of n. Negative or non-integer values are not applicable in this context.

What is the sum of the infinite series 4^n / 2^n?

The infinite series 4^n / 2^n diverges to infinity because the terms grow without bound as n increases.