1.2 Series and Convergence

1.2.1 Definitions

The mathematics in this section is nothing new either — it’s really just sequences in disguise. Given a sequence, one thing we can do is add up all the terms and see what happens. This results in the idea of an (infinite) series:

Definition 1.5: (Series)

Let \((a_n)_{n \in \mathbb{N}}\) be a real sequence. Then \[\sum_{n = 1}^{\infty} a_n\] is called a series for \((a_n)_{n\in\mathbb{N}}\).

So when does this series converge? In other words, when is the object in Definition 1.5 a real number? To answer these questions, we have the following definition:

Definition 1.6: (Series Convergence and Partial Sums)

Let \((a_n)_{n \in \mathbb{N}}\) be a real sequence. Then \(\sum_{n = 1}^{\infty} a_n\) converges if and only if the sequence \((S_N)_{N \in \mathbb{N}}\) converges, where \[S_N:= \sum_{n = 1}^{N} a_n\] is the \(N^{\text{th}}\) partial sum. If \(S_N \to \ell\) as \(N \to \infty\), we define \[\ell = \sum_{n = 1}^{\infty}a_n.\]

1.2.2 Tests for Convergence

Much like with proving sequence convergence, using the definition each time you want to ‘evaluate’ a series can get tedious really quickly. Therefore, we really want a couple of tests which can prove convergence without too much hassle. The first of these tests involves comparing the sizes of two series, and is aptly known as the comparison test.

Theorem 1.4: (Comparison Test)

Let \((a_n)_n\) and \((b_n)_n\) be real sequences, and suppose that there exists a \(M \in \mathbb{N}\) such that \(\lvert a_n \rvert \leq b_n \;\forall n \geq M.\) Then, if \(\sum_{n = 1}^{\infty} b_n\) is convergent, \(\sum_{n = 1}^{\infty} a_n\) is convergent.

Naturally, using this, we can also build a test for divergence to \(\infty\) out of the comparison test too.

Corollary 1.1:

Let \((a_n)_n\) and \((b_n)_n\) be real sequences. If there exists a \(M \in \mathbb{N}\) such that \(0 \leq a_n \leq b_n \; \forall n \geq M\), and \(\sum_{n = 1}^{\infty} a_n\) diverges, then \(\sum_{n = 1}^{\infty} b_n\) diverges.

The second test we look at here is similar to the Growth Factor test (Theorem 1.2), in that it assesses convergence of a series by examining the ratio of successive terms:

Theorem 1.5: (D’Alembert’s Ratio Test)

Let \((a_n)_n\) be a real sequence with \(a_n \neq 0 \; \forall n \in \mathbb{N}\). Suppose \[\lim_{n\to\infty}\frac{\lvert a_{n+1}\rvert}{\lvert a_n\rvert} = r.\] Then:

• If \(0 \leq r < 1\), \(\sum_{n = 1}^{\infty} a_n\) converges.
• If \(r > 1\), then \(\sum_{n = 1}^{\infty} a_n\) diverges.
• If \(r = 1\), the test is inconclusive.

Like the Growth Factor Test, d’Alembert’s test fails if the terms of the series are formed of ratios of polynomials.

The final test presented here is applicable when the terms of the series alternate in sign:

Theorem 1.6: (Leibniz Alternating Series Test)

Suppose \((a_n)_{n\in\mathbb{N}}\) is a decreasing sequence tending to \(0\) as \(n \to \infty\). Then \[\sum_{n=1}^{\infty} (-1)^n a_n\] is a convergent series. Further, the value of this series lies between \(-a_1\) and \(a_2 - a_1\).