Fundamental concepts of infinite series, including convergence, divergence, partial sums, and the crucial first-step check: the n-th Term Test for Divergence. Also covers the essential geometric series.
An infinite series is the sum of the terms of an infinite sequence \(a_n\). It is denoted by \(\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots\).
What does it mean for an infinite series to converge?
A series converges if its sequence of partial sums, \(S_N = \sum_{n=1}^{N} a_n\), approaches a finite limit \(L\) as \(N \to \infty\). This limit \(L\) is called the sum of the series.
What does it mean for an infinite series to diverge?
A series diverges if its sequence of partial sums does not approach a finite limit. This can happen if the partial sums grow to infinity, decrease to negative infinity, or oscillate without approaching a single value.
State the n-th Term Test for Divergence.
For a series \(\sum a_n\), if the limit of its terms does not go to zero (i.e., \(\lim_{n \to \infty} a_n \neq 0\)) or the limit does not exist, then the series must diverge.
If \(\lim_{n \to \infty} a_n = 0\), does the series \(\sum a_n\) necessarily converge?
No. This is a common mistake. If \(\lim_{n \to \infty} a_n = 0\), the test is inconclusive. The series might converge or it might diverge. For example, the harmonic series \(\sum \frac{1}{n}\) diverges, yet its terms approach \(0\).
Apply the n-th Term Test to the series \(\sum_{n=1}^{\infty} \frac{n}{2n+1}\).
We compute the limit: \(\lim_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2}\). Since the limit is not \(0\), the series diverges by the n-th Term Test.
What is the general form of a geometric series?
A geometric series has the form \(\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + \dots\), where \(a\) is the first term and \(r\) is the common ratio.
Under what condition does a geometric series converge?
A geometric series converges if and only if the absolute value of its common ratio \(r\) is less than \(1\). That is, \(|r| < 1\).
What is the formula for the sum of a convergent geometric series \(\sum_{n=0}^{\infty} ar^n\)?
If \(|r| < 1\), the sum is given by the formula \(S = \frac{a}{1-r}\), where \(a\) is the first term.
Does the series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) converge or diverge? If it converges, what is its sum?
This is a geometric series with first term \(a = \frac{1}{2}\) and common ratio \(r = \frac{1}{2}\). Since \(|r| < 1\), it converges. Its sum is \(S = \frac{a}{1-r} = \frac{1/2}{1 - 1/2} = 1\).
What is the harmonic series, and does it converge or diverge?
The harmonic series is \(\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots\). It diverges, even though its terms approach zero. This is a critical counterexample to remember.
