Can you add up infinitely many numbers and still get a finite answer? Sometimes yes, and sometimes no. This chapter gives you the tools to decide which case you are in, and then to turn that knowledge into powerful formulas for functions such as \(e^x\) and \(\ln(1+x)\).
1. Sequences and their limits
A sequence \((a_n)\) is a list of numbers indexed by \(n = 1, 2, 3, \dots\). It converges to \(L\) if \(a_n\) gets as close to \(L\) as you like, and stays that close, once \(n\) is large enough. We write \(\lim_{n\to\infty} a_n = L\). If no such \(L\) exists, the sequence diverges.
Most limits you meet are found by dividing by the highest power of \(n\) and using \(\dfrac{1}{n}\to 0\). Two more facts help: if \(a_n = f(n)\) and \(f(x)\to L\) as \(x\to\infty\), then \(a_n\to L\); and a sequence that is bounded and monotone (always increasing or always decreasing) must converge.
Find \(\lim_{n\to\infty}\dfrac{4n^2+1}{n^2-3n}\). Divide the top and bottom by \(n^2\):
\[\frac{4+\frac{1}{n^2}}{1-\frac{3}{n}}\longrightarrow \frac{4+0}{1-0}=4.\]
The sequence converges to 4.
In the figure above, the terms of \(a_n=\dfrac{2n+1}{n+1}\) climb toward the dashed line \(y=2\). Every term stays below 2, yet the gap shrinks to zero.
2. Series and geometric series
A series is the infinite sum \(\sum_{n=1}^{\infty} a_n\). Its partial sums are \(s_N = a_1 + a_2 + \cdots + a_N\). The series converges to \(S\) when \(s_N \to S\); otherwise it diverges. So a series is just a sequence of partial sums.
For \(a\neq 0\), the series \(\sum_{n=0}^{\infty} a r^n\) converges if and only if \(|r| < 1\), and then
\[\sum_{n=0}^{\infty} a r^n=\frac{a}{1-r}.\]
If \(|r|\ge 1\) it diverges.
The reason is the identity \(s_N = a\dfrac{1-r^{N+1}}{1-r}\): when \(|r| < 1\) the power \(r^{N+1}\) vanishes in the limit.
Compute \(\sum_{n=0}^{\infty} 7(0.8)^n\). Here \(a=7\) and \(r=0.8\), so \(|r| < 1\) and the sum is \(\dfrac{7}{1-0.8}=35\).
The partial sums of \(1+\tfrac12+\tfrac14+\cdots\) shown above creep up to 2 but never pass it.
3. The divergence test and the integral test
If \(\sum a_n\) converges, then \(a_n\to 0\). Equivalently: if \(a_n\) does not tend to 0, then \(\sum a_n\) diverges.
The test only works in one direction. If \(a_n\to 0\) you learn nothing. The harmonic series \(\sum \dfrac1n\) has terms tending to 0, yet it diverges.
Let \(f\) be positive, continuous and decreasing on \([1,\infty)\) with \(a_n=f(n)\). Then \(\sum a_n\) and \(\int_1^{\infty} f(x)\,dx\) either both converge or both diverge. As a consequence, the p-series \(\sum \dfrac{1}{n^p}\) converges exactly when \(p > 1\).
Each rectangle in the figure has area \(\tfrac1n\) and lies above the curve \(y=\tfrac1x\). Their total area is at least \(\int_1^7 \tfrac{dx}{x}=\ln 7\), and \(\ln N\) grows without bound, so the harmonic series diverges (slowly: you need over 12,000 terms to pass 10).
4. Comparison tests
Let \(a_n, b_n > 0\). (1) If \(a_n\le b_n\) and \(\sum b_n\) converges, then \(\sum a_n\) converges. If \(a_n\ge b_n\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges. (2) If \(\lim \dfrac{a_n}{b_n}=c\) with \(0 < c < \infty\), then both series behave the same way.
- Keep only the dominant terms of \(a_n\) for large \(n\).
- Choose a series \(b_n\) you already understand, usually a p-series or geometric series.
- Check the inequality or compute the limit of \(a_n/b_n\).
Does \(\sum \dfrac{1}{n^2+n+3}\) converge? For \(n\ge 1\), \(n^2+n+3 > n^2\), so \(\dfrac{1}{n^2+n+3} < \dfrac{1}{n^2}\). Since \(\sum \dfrac1{n^2}\) is a convergent p-series (\(p=2\)), the given series converges.
5. The ratio and root tests
Let \(L=\lim \left|\dfrac{a_{n+1}}{a_n}\right|\) (ratio test) or \(L=\lim \sqrt[n]{|a_n|}\) (root test). If \(L < 1\) the series converges absolutely; if \(L > 1\) (or \(L=\infty\)) it diverges; if \(L=1\) the test is inconclusive.
These tests shine with factorials, exponentials and \(n\)-th powers, because the messy parts cancel.
For \(\sum \dfrac{5^n}{n!}\):
\[\frac{a_{n+1}}{a_n}=\frac{5^{n+1}}{(n+1)!}\cdot\frac{n!}{5^n}=\frac{5}{n+1}\longrightarrow 0 < 1.\]
The series converges.
6. Alternating series
If \(b_n > 0\), \(b_n\) is decreasing and \(b_n\to 0\), then \(\sum (-1)^{n+1} b_n\) converges. Moreover the error after \(N\) terms satisfies \(|S - s_N|\le b_{N+1}\).
A series converges absolutely if \(\sum |a_n|\) converges. If it converges but not absolutely, it converges conditionally. Absolute convergence always implies convergence.
The partial sums of \(1-\tfrac12+\tfrac13-\cdots\) bounce around \(\ln 2\approx 0.693\), each swing smaller than the last. This series is conditionally convergent, because \(\sum\tfrac1n\) diverges.
7. Power series
A power series centered at \(c\) is \(\sum_{n=0}^{\infty} c_n (x-c)^n\). It converges on an interval around \(c\) of radius \(R\) (possibly 0 or \(\infty\)): inside \((c-R,\,c+R)\) it converges absolutely, outside it diverges, and the endpoints must be tested one by one.
Use the ratio test on the terms to find \(R\). Inside the interval you may differentiate and integrate term by term, which produces new power series from old ones. For instance, from \(\dfrac{1}{1-x}=\sum x^n\) for \(|x| < 1\) you get \(\dfrac{1}{(1-x)^2}=\sum n x^{n-1}\).
8. Taylor series
If \(f\) has derivatives of all orders at \(c\), its Taylor series is
\[\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n.\]
When \(c=0\) it is called a Maclaurin series.
Three Maclaurin series are worth memorizing: \(e^x=\sum \dfrac{x^n}{n!}\) for all \(x\); \(\sin x=\sum (-1)^n\dfrac{x^{2n+1}}{(2n+1)!}\) for all \(x\); and \(\ln(1+x)=\sum_{n\ge1} (-1)^{n+1}\dfrac{x^n}{n}\) for \(-1 < x\le 1\).
Use the first four terms of \(e^x\) at \(x=0.5\):
\[1+0.5+\frac{0.5^2}{2}+\frac{0.5^3}{6}=1+0.5+0.125+0.02083\approx 1.6458.\]
A calculator gives \(e^{0.5}\approx 1.6487\), so the error is about 0.003.
On my home planet we say: to survive a series, ask three questions in order. Do the terms go to zero? Does it look like a p-series or a geometric series? Is there a factorial or an \(n\)-th power? The answer tells you the test to use.
Key takeaways
- A series converges when its sequence of partial sums has a finite limit.
- Geometric series: \(\sum a r^n = \dfrac{a}{1-r}\) when \(|r| < 1\), divergent otherwise.
- If \(a_n\not\to 0\), the series diverges; if \(a_n\to 0\), you still need another test.
- p-series converge exactly when \(p > 1\); the integral and comparison tests extend this to similar series.
- Ratio and root tests: \(L < 1\) converges, \(L > 1\) diverges, \(L=1\) tells you nothing.
- Alternating series with decreasing terms tending to 0 converge, and the error is at most the first omitted term.
- Power series have a radius of convergence; test the endpoints separately.
- The Taylor series of \(f\) at \(c\) has coefficients \(\dfrac{f^{(n)}(c)}{n!}\).
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