Sequences & series
Sequences describe ordered values. Series add those values. The central question is whether an infinite process approaches a finite limit — and, if it does, how fast and why.
1. Sequences and convergence
A sequence is a function whose input is a positive integer: \(a_1,a_2,a_3,\ldots\). We say \(a_n\) converges to \(L\) if its terms can be made arbitrarily close to \(L\) by taking \(n\) sufficiently large.
\(a_n=1/n\to0\).
\(a_n=n\) grows without bound.
\(a_n=(-1)^n\) has no single limit.
Useful facts: sums, products and quotients of convergent sequences behave like ordinary limits when denominators stay away from zero. A bounded monotone sequence always converges.
2. Infinite series and partial sums
An infinite series is defined through its partial sums.
Absolute convergence means \(\sum |a_n|\) converges. Absolute convergence guarantees convergence of \(\sum a_n\), while some alternating series converge only conditionally.
3. Benchmark series worth knowing
Geometric series
If \(|r|\ge1\), the geometric series diverges.
p-series
The harmonic series corresponds to \(p=1\) and diverges.
Telescoping series
Cancellation can reduce a long partial sum to just a few boundary terms. For example,
4. Choosing a convergence test
| Test | Best used when | Decision |
|---|---|---|
| nth-term test | First check for every series | If \(a_n\not\to0\), the series diverges. |
| Comparison | Positive terms resemble a known benchmark | Compare with convergent/divergent positive series. |
| Limit comparison | Same asymptotic size as a benchmark | If \(a_n/b_n\to c\in(0,\infty)\), both behave alike. |
| Ratio test | Factorials or exponentials | For \(L=\lim|a_{n+1}/a_n|\): converge if \(L<1\), diverge if \(L>1\). |
| Root test | Terms raised to the nth power | For \(L=\limsup\sqrt[n]{|a_n|}\): same threshold as ratio test. |
| Alternating-series test | Signs alternate | If magnitudes decrease to zero, the series converges. |
| Integral test | \(a_n=f(n)\) with positive decreasing \(f\) | \(\sum a_n\) and \(\int f\) converge or diverge together. |
5. Worked examples
A — geometric series
Evaluate \(\sum_{n=0}^{\infty}3(1/4)^n\).
B — limit comparison
Consider \(\sum_{n=1}^{\infty}\frac{3n+2}{n^2+1}\). Compare with \(1/n\):
Because the harmonic series diverges, the given series diverges too.
C — ratio test
For \(\sum n!/3^n\),
so the series diverges.
D — alternating error bound
For an alternating series with decreasing terms \(b_n\to0\), the error after \(N\) terms satisfies
6. Power series and Taylor series
A power series centered at \(c\) has the form
There is a radius of convergence \(R\): the series converges absolutely for \(|x-c|
Taylor series
If a function is represented by its Taylor series near \(c\), then
Important Maclaurin examples include
A Taylor polynomial is a finite approximation; the remainder controls its error.
7. Interactive partial-sum explorer
Compare the first \(N\) partial sums of a geometric series, a p-series or an alternating harmonic series.
8. Common mistakes
- Assuming \(a_n\to0\) is enough for \(\sum a_n\) to converge.
- Using the ratio or root test and forcing a conclusion when the limit equals \(1\); those tests are then inconclusive.
- Forgetting to test endpoints of a power series separately.
- Confusing sequence convergence with series convergence.
- Claiming conditional convergence without first checking that the alternating conditions hold.
9. Practice
- Determine whether \(a_n=(2n+1)/(n+4)\) converges and find its limit.
- Evaluate \(\sum_{n=0}^{\infty}5(1/3)^n\).
- Determine whether \(\sum 1/n^{3/2}\) converges.
- Use the ratio test on \(\sum 2^n/n!\).
- Find the radius of convergence of \(\sum_{n=1}^{\infty}(x-2)^n/n\).
Solutions
1) \(2\). 2) \(15/2\). 3) Converges because \(p=3/2>1\). 4) Ratio tends to \(0\), so it converges absolutely. 5) \(R=1\); endpoints require separate analysis.