Understanding and Calculating the Sum to Infinity of Infinite Series

When dealing with infinite series, particularly geometric or convergent series, finding the sum to infinity is a common objective. This article provides an overview of various types of infinite series, their convergence tests, and methods for calculating the sum to infinity where applicable.

Geometric Series

A geometric series is a sequence of the form S a ar ar^2 ar^3 ldots where a is the first term and r is the common ratio. The sum to infinity of a geometric series converges if the absolute value of r is less than 1 (r 1). The formula for the sum is:

S frac{a}{1 - r}

Example: Sum to Infinity of a Geometric Series

Consider the series: 2, 2.5, 2.5^2, ldots a 2 and r 0.5

The sum to infinity is calculated as:

S frac{2}{1 - 0.5} frac{2}{0.5} 4

Convergent Series

For non-geometric series, the series must converge for the sum to have a finite value. Various tests can be used to check for convergence:

Ratio Test – Determines the convergence of a series based on the ratio of successive terms. Root Test – Determines the convergence of a series based on the nth root of the nth term. Comparison Test – Compares the given series to another series whose convergence is known. Integral Test – Applies if the series can be expressed as the integral of a function.

Examples of series that converge include:

Example: Sum of the Series sum_{n1}^{infty} frac{1}{n^2}

The series converges, and its sum is known to be:

frac{pi^2}{6}

Divergent Series

If a series diverges, it does not have a sum to infinity in the traditional sense. Such series are not converging to a finite value as the number of terms increases.

Using Finite Differences to Find a Formula for a Sequence

To find a formula for a sequence, such as the sequence 5, 7, 11, 17, ldots, we can use the method of finite differences. For the sequence given, the first row of differences yields 2, 4, 6, and the second row of differences yields a constant sequence 2, 2. Therefore, our formula is a second-degree polynomial:

a_k ak^2 - bk - c

By making it easy to work with, we set a_3 5, a_4 7, etc., so our indices match with our powers of x. We have three unknown coefficients to solve for, so we plug in three values of our sequence:

5 9a - 3b - c

7 16a - 4b - c

11 25a - 5b - c

Solving these equations, we obtain a 1, b -5, c 11, and the formula:

a_k k^2 - 5k - 11

Our sum now becomes:

sum_{k3}^{infty} (-1)^{k-1}(k^2 - 5k - 11)x^k

We utilize geometric series formulas to find:

sum_{k0}^{infty} (-1)^{k-1}(k^2 - 5k - 11)x^k sum_{k0}^{infty} (-1)^{k-1}k^2x^k - sum_{k0}^{infty} (-1)^k5kx^k - sum_{k0}^{infty} (-1)^{k-1}11x^k

We refer to these as [1], [2], and [3] respectively. These are split and solved using known geometric series results:

[3] frac{-11}{1 - x}

[2] frac{-5x}{1 - x^2}

[1] frac{x - 1}{1 - x^3}

Substituting everything in and subtracting the first three terms, we find:

sum_{k3}^{infty} (-1)^{k-1}(k^2 - 5k - 11)x^k frac{x - 1}{1 - x^3} - frac{5x}{1 - x^2} - frac{11}{1 - x} - (-11)(-7x) - (5)(11x^2)

After a large amount of tedious algebra, we arrive at our final answer:

frac{x^3 - 5x^2 - 8x - 5}{1 - x^3}

Note: This series only converges when x 1 since the various geometric series used only converge when x 1.