Sequences and series | Improve Society

A sequence, indicated by u₁, u₂, …or brief by $\langle u_n \rangle$ , is a function defined on the set of natural numbers. The sequence is said to have the limit l or to converge to l, if given any ε > 0 there exists a number N > 0 such that |u_n – l| N, and in such case it is described $\lim\limits_{n \rightarrow \infty} u_n =l$ . If the sequence does not converge, it’s called that it diverges.

Consider the sequence u₁, u₁ + u₂, u₁ + u₂ + u₃, … or S₁, S₂, S₃, … where S_n = u₁ + u₂ + … + u_n. It’s called $\langle S_n \rangle$ the sequence of partial sums of the sequence $\langle u_n \rangle$ . The symbol

$\displaystyle u_1 + u_2 + u_3 + \cdots$ or $\displaystyle \sum_{n=1}^{\infty}u_n$ or briefly $\displaystyle \sum u_n$

is defined as synonymous with $\langle S_n \rangle$ and is called an infinite series. This series will converge or diverge according as $\langle S_n \rangle$ converges or diverges. If it converges to S it’s called S as the sum of the series.

The following are some important theorems concerning infinite series.

The series $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^p}$ converges if p > 1 and diverges if p ≤ 1.

If ∑|u_n| converges and |v_n| ≤ |u_n|, then ∑|v_n| converges.

If ∑|u_n| converges, then ∑u_n converges.

If ∑|u_n| diverges and v_n ≥ |u_n|, then ∑v_n diverges.

The series ∑|u_n|, where |u_n| = f(n) ≥ 0, converges or diverges according as $\displaystyle \int_{1}^{\infty}f(x)dx = \lim\limits_{M \rightarrow \infty}\int_{1}^{M}f(x)dx$ exists or does not exist. This theorem is often called the integral test.

The series ∑|u_n| diverges if $\displaystyle \lim\limits_{n \rightarrow \infty}|u_n| \neq 0$ . However, if $\displaystyle \lim\limits_{n \rightarrow \infty}|u_n| = 0$ the series may or may not converge.

Suppose that $\displaystyle \lim\limits_{n \rightarrow \infty}\left|\frac{u_{n+1}}{n_n}\right| = r$ . Then the series ∑u_n converges (absolutely) if r 1. If r = 1, no conclusion can be drawn. This theorem is often referred to as the ratio test.