Unit 10: Infinite Sums and Series (BC-only)
This BC-only unit is about representing functions and numbers through infinitely many terms, and then deciding when those infinite processes make sense.
Sequences
A sequence is a function whose domain is the positive integers:
\[a_1, a_2, a_3, \dots\]We write
\[\lim_{n \to \infty} a_n = L\]if the terms approach \(L\).
If \(\sum a_n\) converges, then necessarily \(a_n \to 0\). The converse is false.
Infinite series
A series is the sum
\[\sum_{n=1}^{\infty} a_n.\]Its convergence is defined by the sequence of partial sums:
\[S_N = \sum_{n=1}^{N} a_n.\]Geometric series
\[\sum_{n=0}^{\infty} ar^n\]converges when \(\lvert r \rvert < 1\) and then
\[\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}.\]Harmonic and p-series
The harmonic series
\[\sum_{n=1}^{\infty} \frac{1}{n}\]diverges.
The p-series
\[\sum_{n=1}^{\infty} \frac{1}{n^p}\]converges if and only if \(p>1\).
Integral Test
If \(f(x)\) is positive, continuous, and decreasing for large \(x\) with \(f(n)=a_n\), then
\[\sum a_n\]and
\[\int f(x)\,dx\]either both converge or both diverge.
Comparison tests
Direct comparison:
- if \(0 \le a_n \le b_n\) and \(\sum b_n\) converges, then \(\sum a_n\) converges,
- if \(0 \le b_n \le a_n\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.
Limit comparison:
If
\[\lim_{n \to \infty} \frac{a_n}{b_n} = c\]with \(0<c<\infty\), then \(\sum a_n\) and \(\sum b_n\) behave the same.
Alternating series
An alternating series often has the form
\[\sum_{n=1}^{\infty} (-1)^n b_n\]or
\[\sum_{n=1}^{\infty} (-1)^{n+1} b_n\]with \(b_n > 0\).
The Alternating Series Test says the series converges if:
- \(b_n\) decreases eventually,
- \(b_n \to 0\).
Absolute vs conditional convergence
If
\[\sum \lvert a_n \rvert\]converges, then \(\sum a_n\) converges absolutely.
If \(\sum a_n\) converges but \(\sum \lvert a_n \rvert\) diverges, the convergence is conditional.
Ratio and root tests
Ratio Test:
\[L = \lim_{n \to \infty} \left\lvert\frac{a_{n+1}}{a_n}\right\rvert\]Root Test:
\[L = \lim_{n \to \infty} \sqrt[n]{\lvert a_n \rvert}\]In either test:
- if \(L<1\), converge absolutely,
- if \(L>1\) or infinite, diverge,
- if \(L=1\), inconclusive.
nth-term test for divergence
If
\[\lim_{n \to \infty} a_n \ne 0\]or the limit does not exist, then
\[\sum a_n\]diverges.
Power series
A power series centered at \(c\) has form
\[\sum_{n=0}^{\infty} a_n(x-c)^n.\]There is a radius of convergence \(R\):
- converges for \(\lvert x-c \rvert<R\),
- diverges for \(\lvert x-c \rvert>R\),
- endpoints must be checked separately.
[Image Placeholder: number line showing center, radius, and endpoint testing]
Taylor and Maclaurin series
The Taylor series of \(f\) centered at \(c\) is
\[\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n.\]Maclaurin series is the special case \(c=0\).
Core series to memorize:
\[\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n, \qquad \lvert x \rvert<1\] \[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\] \[\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\] \[\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}\]Taylor polynomial and error
The \(n\)th Taylor polynomial is the finite truncation:
\[T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!}(x-c)^k.\]For alternating Maclaurin series with decreasing term magnitudes, the truncation error is at most the first omitted term in absolute value.
Common mistakes
- Forgetting that \(a_n \to 0\) is necessary but not sufficient.
- Using a convergence test whose hypotheses do not apply.
- Stopping after finding the radius of convergence without testing endpoints.
- Mixing up absolute and conditional convergence.