What Is An Infinite Series?

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An infinite series is a sum of the form

a_1 + a_2 + a_3 + a_4 + \dots

where the a_i are a sequence of numbers. In other words, an infinite series (usually we just say “series”) is the sum of terms of a sequence. Of course, it’s ridiculous to consider sums like 1+2+3+4+5+\dots or 1+1+1+1+\dots, and we say that such series diverge. However, some series, such as

\displaystyle 1 + \frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\dots

we can assign a meaningful sum to.

Series are so important that we have a special notation for them: A Greek capital sigma with some special things written below and above.

\displaystyle \sum_{i=1}^{\infty} a_i

This is read “the sum from i equals 1 to infinity of a_i“. It’s a recipe for writing down a sum, which tells us to start at i=1, write down a_1, then go on to i=2, add a_2, then go on to i=3, and so on… In other words,

\displaystyle \sum_{i=1}^{\infty} a_i = a_1+a_2+a_3+\dots.

Notice that the i here is a dummy variable, which means that we could just as easily use j or some other letter:

\displaystyle \sum_{i=1}^{\infty} a_i = \sum_{n=1}^{\infty} a_n

Furthermore, we can change the starting value of the variable we’re summing over and change the index of the sequence we’re summing. As an example,

\displaystyle \sum_{i=1}^{\infty} a_i = \sum_{i=0}^{\infty} a_{i+1}

To convince yourself that this is true, write down what the sum on the left means and then write down what the sum on the right means.

Infinite series arise often in calculus, but they’re also important in other branches of science. In physics, many important functions are defined through series. In computer science there are many important sequences of numbers which can be more easily explored using special series called “generating series.”