Infinite Series

An infinite sum is the sum of an infinite sequence of terms $a_1 + a_2 + a_3 + a_4 + a_5 + ... + a_n$.

The goal is to come to understand the meaning of such an infinite sum and to develop methods to calculate it.

Note: See MIT's Calculus Revisited lecture, Many vs Infinite, which introduces infinite series. It has an excellent definition and description of just how profound infinity is.

Notation

Conventionally, infinite series use sigma notation. These are example of acceptable index variables.

$$\sum_{n=1}^{\infty} a_n, \quad \quad \sum_{k=1}^{\infty} b_k, \quad \text{ or } \quad \sum a_n$$

The final one has no index variable, which is a useful shortcut when summation from $1$ to $\infty$ is understood.

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Calculating Infinite Sums

The sum of a series = the limit of the sequence of partial sums.—_i.e.:_

$$\sum a_n = \lim_{n \to \infty} S_n \quad \text{ where } S_n = \text{ the sequence of partial sums }$$

Since infinite sums always have more terms, no matter where you stop counting, the method for calculating the sum of an infinite series is:

1. Evaluate the result of adding the first $n$ terms, then stopping. The sum of the first $n$ terms is called the Nth Partial Sum.
2. Take the limit as $n \to \infty$ and see if the limit exists (i.e. the sum approaches a finite number.

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Convergence vs Divergence

Convergence

The series converges if the second step above results in a finite limit.

Divergence

If the limit isn't finite, the series diverges:

• If the limit goes to infinity, the series is said to diverge to infinity.
• If the series has no limit, because it oscillates between two or more values, then it simply diverges.

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Partial Sums

The sum of the first $n$ terms

$$S_n = a_1 + a_2 + a_3 + a_4 + a_5 + ... + a_n$$

is an ordinary finite sum you get just through simple addition. It's called the nth partial sum.

For instance, the nth partial sum of the sequence $\{a_n\}$ apparently approaches the number 2:

$$\{a_n\} = 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, ..., a_n$$$$S_n = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + ... s_n$$$$\lim_{n\to\infty} \{S_n\} = 2$$